RADICALS AND SOLVING QUADRATIC EQUATIONS

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1 RADICALS AND SOLVING QUADRATIC EQUATIONS Evaluate Roots Overview of Objectives, studets should be able to:. Evaluate roots a. Siplify expressios of the for a b. Siplify expressios of the for a. Evaluate higher order eve ad odd roots a. Siplify expressios of the for a Objectives: Siplify expressios of the for a Mai Overarchig Questios:. How do you siplify radical expressios with variables ad costats?. What is the differece betwee the rules for eve roots ad odd roots? Activities ad Questios to ask studets: If ecessary review the basics of siplifyig square root expressios foud o pgs of the MAT 000/00 guidebook. I short: what is 9? What is? After a few ore exaples of this kid, have studets draw to the coclusio that a = a. Ask studets to give a verbal descriptio of this rule (i.e. if we take the square root of a squared quatity, the result is the origial quatity ot squared. ) To the exted the rule to egatives, go back to the origial exaple, ad ask studets what ( ) is. If they assue there is NO SOLUTION, ask the to follow order of operatios to siplify. What is the result? Was the result the sae as the origial quatity beig squared i the radical? What happeed? Is the result siilar to the origial quatity beig squared? Does our origial rule a = a hold? Why or why ot? Repeat this type of exaple a couple of ties. What patter do you otice? What happes to the quatity beig squared iside the radical each tie? What is the sig of the result? Oce studets otice the result is always the positive of the uber beig squared iside the The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

2 Siplify expressios of the for a Siplify expressios of the for a radical, ask the to thik about what operator we have studied that always akes thigs positive. They ay cosider usig a egative sig, but reid the we wat to use the sae operator for all ubers. Have studets draw to the ew coclusio that a = a. If studets eed or would like to practice at this poit give a few probles fro the followig (easiest to hardest): ( 0), ( x ), x + x + 6 9x, 6 If ecessary review the basics of siplifyig cube root expressios foud o pgs of the MAT 000/00 guidebook. I short: what is 8? What is? After a few ore exaples of this kid, have studets draw to the coclusio that a = a. Ask studets to give a verbal descriptio of this rule (i.e. if we take the cube root of a cubed quatity, the result is the origial quatity ot cubed. ) Hopefully, the studets will ask about the egative cases. Go back to the origial exaple, ad ask studets what ( ) is. If they assue there is NO SOLUTION, ask the to follow order of operatios to siplify. What is the result? Was the result the sae as the origial quatity beig cubed i the radical? What happeed? Is the result siilar to the origial quatity beig squared? Does our origial rule a = a hold? Why or why ot? Repeat this type of exaple a couple of ties. What patter do you otice? What happes to the quatity beig squared iside the radical each tie? What does the sig of the aswer deped o? How is this rule differet fro the square root case? If studets eed or would like to practice at this poit give a few probles fro the followig (easiest to hardest): ( 00), ( x +), 5x We wat the studets to establish a rule for geeral th degree roots. Studets should have oticed that with square roots we eeded a absolute value bar to esure variable expressios were always positive. For cube roots we should ot iclude the bars because egatives are okay. What about th roots? Studets have ot bee itroduced to roots above the cube root, so ask the what they thik a radical does. If they are usure have the cosider how The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

3 square roots ad cube roots operate ad draw a coclusio about th roots ad higher. If ecessary give the the exaple that = 6 ad the 6 =. Have the state the process i words. Repeat this process for 5 th roots ad higher. Ask studets to use the process fro the previous two discussios to develop the rules for 5 5 what a = ad a =. If ecessary, give ueric exaples to begi the discussio but the aalysis should be siilar. Have studets draw to the coclusio that a = a ad 5 5 a = Ask the studets if they see a patter. Ca they geeralize these rules to all roots? What sees to be happeig for eve roots? Odd roots? Have studets draw to the coclusio that if is eve: a = a Have studets draw to the coclusio that if is odd: a = a For practice use probles that are of a siilar type give above. a Multiply Radical Expressios Overview of Objectives, studets should be able to:. Use the product rule to ultiply radicals. Use factorig ad the product rule to siplify radicals. Multiply radical expressios ad the siplify. Multiply radical expressios with ore tha oe ter. 5. Use polyoial special products to ultiply radicals Objectives: Mai Overarchig Questios:. How do you kow whe radical expressios ca be cobied through ultiplicatio?. How ca you decide which rules to use whe siplifyig radical expressios? Activities ad Questios to ask studets: Use the product rule to ultiply radicals The product rule has already bee established for square roots o pg 77 of the MAT 000/00 discussio. However, give the iportace of this rule i both ultiplicatio AND siplificatio, it is probably worth a secod discussio. The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

4 Use factorig ad the product rule to siplify radicals Ask studets to evaluate: 9. Ask studets to evaluate: 9. Do you see a relatioship? Have studets work several pairs of exaples of this type. Does a patter exist? What is a b? What about a b? Reid studets to thik about what the square root process is. Do you otice a patter? Have studets draw the coclusio that: a b = a b Have studets work several exaples with higher order roots to establish: a b = a b. Ask studets if they see ay liitatios to the rule. Ca we ultiply ay radicals? What requireets does the rule establish? If studets have trouble, ask the if we ca apply the rule to this proble: x x. Why or why ot? Give studets a few proble to ultiply (they ay ot kow how to siplify; this will be itroduced i the ext cocept) This cocept was also itroduced i MAT 00 but i sipler types of probles. As a review: How would you siplify 8 = 9 usig the product rule? Describe the process. Whe breakig up 8 to 9, what process are you usig? Why would you choose to factor 8 ito 9 istead of aother choice like 6? How would you kow how to factor? Suarize the siplifyig process. Studets should see the goal is to factor the radicad such that oe of the factors is the greatest possible perfect square. How about? How does the o the outside chage the process? Have studets work several ueric exaples of this type as a review. 6 Start with a exaple like x. Ask studets to siplify the radical. Give several exaples with eve expoets for studets to work. How is this like what we have doe before? If ecessary reid studets of what rules have bee established so far for radicals. 7 6 What about x = x x? Do you see a patter or process to if the expoet is odd? Give studets several ore exaples to work. If ecessary, give a suary of probles of the followig types (easiest to hardest): 50, The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

5 7x, 00 y The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret. 9 x, x ), x 6x + 6 ( + Now, the factorig ad siplifyig process will be exteded to higher order roots. Whe siplifyig square roots we wated to factor the radicad ito the greatest possible perfect square. Ask studets how they would hadle cube roots. What should we factor our radicad ito? Let studets discuss soe ueric higher order exaples: 5, 6 Start with a exaple like x What about x 6 So, what quatity cubed gives us Next, x 9. What is the result?. Studets ay have soe trouble with this. Ask the what the cube root does. 6 x?. What is the result? Do you see a patter with the cube root? How does the power of radicad chage whe takig the cube root? Do you see a patter? Next, give studets a siple exaple to siplify: x = x x. Ask studets to explai a way to siplify cube roots where the power of the radicad is ot a ultiple of three. Give a few exaples to work: 0 y, 5 ( x + y), 9 7 x y Give the process for siplifyig square roots ad cube roots, ask studets to cojecture how they would hadle higher order roots. If they have trouble, say: For square roots we divided eve powers i the radicad by to take the root, for cube roots we divided powers that are ultiples of by, what do you thik we would do with th, 5 th, etc roots? What ust the powers i the radicad be ultiples of i case? This is usually a difficult cocept for studets to see sice there are so ay differet types of probles, so at this poit, ask the to suarize the process for siplifyig roots i geeral ad add specifics for each type of root. Have studets use their process to work the followig types of probles: 5 7 y, 96x Multiply radical expressios ad the siplify This cocept is a cobiatio of the ultiplicatio ad siplifyig cocepts just itroduced. To facilitate the discussio give studets a proble to look at: 50xy xy. Ask the to ultiply ad siplifyig the expressio. The ask the to suarize the process they used i words, highlightig the ajor steps (i.e. cobie the two radicals through ultiplicatio, factor each quatity i the ew radicad, take the square root, ad the recobie ay reaiig radicals ito oe radical). 50xy xy = 0 x y y Next, give the the sae proble, with the added coditio that all variables are cosidered to be positive quatities. May of the will thik a ajor chage is ecessary, but ask the

6 Multiply radical expressios with ore tha oe ter. to look at their fial aswer. If all variables are positive, what operator could be reoved? Why could the absolute value bars we reoved? 50xy xy = 0xy y. Fro ow o, all variables will be assued to be positive reovig the eed to iclude absolute value bars. Give studets several probles to practice with of varyig difficulty level (square roots, cube roots, ad higher order roots). Iclude at least oe proble where the product rule caot be used such as x x Ask studets to ultiply: 6x ( x ). Which previously leared ultiplicatio techique did you eed to solve the proble? If studets are usure, give the the proble 6x( x ) to siplify. Now go back to the origial proble. Give a worksheet with several ultiplicatio probles (FOIL). Ask studets to write dow the previously leared ultiplicatio techique they eeded to solve. A + B, ( A B), ( A + B)( A B) Give studets a radical exaple of each patter type to siplify. As a chage of pace, ask studets to suarize how old techiques (like factorig, expoets, ad patter ultiplicatio) are ow beig applied to these ore difficult probles. Use polyoial special products to ultiply radicals As a review ask studets to siplify: ( ) Add ad Subtract Radical Expressios Overview of Objectives, studets should be able to:. Add ad subtract like radical expressios. Add ad subtract radical expressios that require siplificatio Mai Overarchig Questios:. How do you kow whe to add or subtract radicals?. How do you kow whe a radical expressio is fully siplified? Objectives: Activities ad Questios to ask studets: The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

7 Add ad subtract like radical expressios Studets have leared this cocept i MAT 00, but as before a quick review discussio will be useful. Ask studets how they would siplify x + x = 5x. Where did the 5 coe fro? Why did t x becoe x? If studets have difficulty, ask What is apples plus apples? Is it 5 apples (x), or 5 tageries (x )? Now give siple radical exaple: What is Ask studets to cosider 5 is x or the apple as i the previous exaples. Write dow the process you used to add the radicals. Give several ore siple exaples What about 9 + 6? Ca you siplify like you did before? Why or why ot? If studets have trouble ad guess = 5 = 5 ask the if there was aother way to fid the aswer = + = 7. Why did the first way ot work? What if variables are icluded? What is x + x. If studets have trouble, phrase it this way: What is square root of x s plus square root of x s. If they say five, ask We have five of what? Reid the of what did ot chage i the origial exaple. How ca you tell that square roots caot be added or subtracted together? Write dow several exaples where they ca ad where they caot be added or subtracted. Ca the sae rule hold for cube roots ad higher roots? Give studets several probles to try with the process o (with costats ad variables) such as 9 7 7, 6 5 x x, ad 9x 8 + x Add ad subtract radical expressios that require siplificatio If + 9 caot be cobied together, is there aother way to siplify? Have studets work several exaples of this type. What process ca you use to siplify? (siplify ad the add or subtract) What about +? How ca you siplify each radical first? Have you doe this before? What rule or process ca you use to help you siplify? How does the o the outside of the radical chage the process? Write dow the process you used to solve the proble. How about 5 x + 8 x? Write dow the process you used to solve. Give a worksheet with several probles of varyig difficulty (with differet roots ad variables) such as: The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

8 5 y 8x xy + ad x y + 5x xy Ask studets to talk through their strategy o how to solve. Is there ore tha oe way to perfor the exaples? Is it correct to say there are ay ways to siplify? Why? Divide Radical Expressios Overview of Objectives, studets should be able to:. Use the quotiet rule to siplify radical expressios. Use the quotiet rule to divide radical expressios. Mai Overarchig Questios:. How do you divide radicals?. How do you kow whe a radical expressio is fully siplified? Objectives: Activities ad Questios to ask studets: Use the quotiet rule to siplify radical expressios Ask studets to evaluate: 6 6 Ask studets to evaluate:. Do you see a relatioship? Have studets work several pairs of exaples of this type. Does a patter exist? a a What is? What about? b b Do you otice a patter? a a Have studets draw the coclusio that: = b b The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

9 Ask studets to show that the rule ca be exteded to: a = b Give studets a few exaples to deostrate the quotiet rule such as: a b 50x 7y 8 y x ad 7 Use the quotiet rule to divide radical expressios a a Ask studets to rewrite the quotiet rule for radicals: = b b How could you use the rule to divide two radicals? Write the process you would use. Ask studets to divide: Ask studets to divide: the proble? 0. Write the process you used to siplify. 0 50x y 5 x. Which previously leared techique did you eed to solve Ratioalize Deoiators Overview of Objectives, studets should be able to:. Ratioalize deoiators with oe ter a. Square root deoiators b. Higher idex root deoiators. Ratioalize deoiators with ore tha oe ter Mai Overarchig Questios:. How do you ratioalize the deoiator of radical expressio?. How do you kow whe a radical expressio is fully siplified? Objectives: Activities ad Questios to ask studets: The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

10 Ratioalize deoiators with oe ter o Square root deoiators o Higher idex root deoiators Ask studets to siplify: 5 5 = 5. Do you otice a patter? Give several ore exaples of this type. Ask studets to divide: =. What is differet about this proble vs. other divisio 5 5 probles? How could you write a equivalet fractio that has o radical i the deoiator? Ask studets to thik about how they ca write equivalet fractios whe addig ad subtractio fractios. Write the process ad operatio you would use. What could you ultiply by to achieve your goal? Write dow your process. Tell studets this process is called ratioalizig the deoiator x Give a exaple: ratioalize the deoiator of. Write dow your steps. x What about for higher order roots? Ask studets how they would ratioalize:. Most of the will assue they oly eed to ultiply by the as they did i the previous process. However this will ot work because the product gives us a radical ter that caot be copletely siplified dow. Reid the that the goal of ratioalizig the deoiator is to reove the radical fro the deoiator. Ask the to figure out what they could ultiply by to get rid of the radical. If they have trouble, ask the to figure out what ubers they ca take the cube root of icely, i.e. =, 8 =, etc. Ask the how they would trasfor to 8 through ultiplicatio. What do we eed to ultiply by? Have the coplete the exaple ad write dow a geeral process to ratioalize deoiators with higher order roots. Give a couple of exaples for the studets to work with higher order 0 5 roots such as:, 5 6 x 7 x y Ratioalize deoiators with ore tha oe ter Ask studets to siplify: ( + 5)( 5). Do you otice a patter? Give several ore exaples of this type. What happes to the radical ters? How could you use this patter to ratioalize the deoiator of:. What could you + 5 ultiply by to reove the radical ter? Write the process you would use. The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

11 Tell studets that if a expressio is a + b the its cojugate is a b or vice versa. Give a exaple: ratioalize the deoiator of. x What is the cojugate of the deoiator? Write dow your steps to solve. Ratioal Expoets Overview of Objectives, studets should be able to:. Use the defiitio of a. Use the defiitio of a. Use the defiitio of a. Siplify expressios with ratioal expoets 5. Siplify radical expressios usig ratioal expoets Objectives: Mai Overarchig Questios:. How do you siplify expressios with ratioal expoets?. How ca you use ratioal expoets to siplify radical expressios?. How do you kow whe your expressio is fully siplified? Activities ad Questios to ask studets: Use the defiitio of a To otivate the discussio, ask studets to cosider 8. We wat to figure out what this quatity is, so let s set a variable equal to this ukow quatity: x = 8. Although studets have ot solved equatios like this before, ask the to thik about what they could do to both sides of the equatio to get rid of the fractioal power. Soe ight cosider ultiplicatio or divisio, but reid the to thik about their expoet rules. It wo t be obvious to cube both sides, so if ecessary give a siple exaple. The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

12 8 Oce studets are o board to cube both sides ask studets to siplify: x = Now, x = 8 = 8. With this is id, ask studets what they would observe x to be (). Reid the of the origial goal, we wated to kow what x = 8 8 was. We ow have =. So what does the fractioal power do? Ask studets to the geeralize what eas. Have studets draw the coclusio that a = a. Later o we will be siplifyig these ratioal expoetial expressios, so reid studets to cosider all the expoet rules they have leared. It ight be beeficial to have studets list out all the expoetial rules they reeber. Ask studets to rewrite the followig expressios as radicals ad siplify where a 5 possible: ( 00), ( 8), (xy) Use the defiitio of a Ask studets to cosider what a Give soe additioal probles to go the other way by rewritig as a fractioal power: 0, 8 x y is, keepig i id that a = a. What is the differece betwee a ad a? What operatio would you eed to do to chage to a. Soe will say ultiply by, but reid the we are dealig with expoets. What do we have to do to ultiply powers? Oce studets realize we eed raise a to the th power, ask the to do this to both sides of the equatio: a = a, a = ( a ), a = ( a ). Tell studets that we ca iterchage the power o the outside of the radical ad ove it to the iside: a ( ) = a. Have studets draw to the coclusio that a = ( a ) ad a ( ) = a. Ask studets if they see a patter or a easier way to reeber what part of the fractioal power represets the root ad what part represets the expoet. Have the geeralize the rule. a The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

13 Give studets the exaple 000 ad ask the to rewrite this as a radical expressio. Sice there are two ways to rewrite these expressios, how do you kow which way is better? Give studets aother exaple like ad ask the to rewrite as a radical expressio. Which of the two ways is better? Ca you geeralize whe to use oe way vs. the other? Give a few ore arithetic probles to work like the two above, additioally have studets practice goig back ad forth betwee radicals ad fractioal expoets. Make sure to give soe where the etire expressio cotais the fractioal power, like ( xy) 7 ad soe where 7 oly parts of the expressio are uder the power like xy Use the defiitio of a This rule will be uch easier to derive. Ask studets to recall what a is. How could we apply this rule to fid a? Have studets coclude that a =. a Does a egative expoet ea we will get a egative uber? Have studets work probles siilar to the last cocept but with egative expoets. Siplify expressios with ratioal expoets At this poit, we will be takig ore coplicated ratioal expoet probles ad siplifyig, so if this has t bee doe, ask studets to write dow all expoetial properties (rules) they reeber. Have the suarize the properties for the class. Reid the that ratioal expoets will use all these properties. Ask studets to also cosider whe a expressio is copletely siplified. Have studets work several probles of varyig degree such as 5 5, x x, 6 y, x, ad x y. For each proble, have studets suarize which expoetial properties they used ad the ordered i which they siplified. Is there ore tha oe way to siplify? Siplify radical expressios usig ratioal expoets This last sectio will apply all the properties of this sectio to siplify radical expressios. The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret. Ask studets how they ight siplify 0 x 5 usig ratioal expoets. Reid the to put the fial aswer back i ters of a radical if a fractioal power reais. Ask studets if they y y 5 8,

14 could have siplified the expressio without ratioal expoets. Allow studets to practice this cocept by givig several exaples to work out. I these probles, it s iportat to write out (with a phrase) what property they are usig at each step ad to keep their work i order. Have the copare ot oly fial aswers, but the order ad properties they used. If soeoe gets the sae aswer, does it ea he or she copleted the proble i the exact aer you did? Hopefully studets will see there are ay ways to siplify these expressios. Radical Equatios Overview of Objectives, studets should be able to:. Solve radical equatios by usig the squarig property of equality. Solve radical equatios by usig the squarig property of equality twice. Objectives:. Solve radical equatios by usig the squarig property of equality Mai Overarchig Questios:. How do you solve radical equatios?. How do you check your solutios are correct?. What is a extraeous solutio? Activities ad Questios to ask studets: Give studets a siple exaple: if x =. What is x =? What about if x =, x =? Do you see a patter? What operatio are we perforig o both sides of the equatio? Have studets draw the coclusio that if a = b the a = b (squarig priciple) Now ask studets how they would solve: x =. If studets just observe the aswer is 6, ask the how they would solve the equatio usig the squarig priciple. Does it atch the solutio you observed? How ca you check your solutio is correct? Write dow the process you used to solve ad check your aswer. How do you solve x =? What happes if you use the squarig priciple? How could you check that 6 is ot the solutio? Metio that solutios that do ot work i a equatio but that are the result of a algebraic ethod are called extraeous solutios Give aother exaple siilar to this oe. Do you see a patter? How could you predict there would be o solutio? The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

15 Solve radical equatios by usig the squarig property of equality twice. How would you solve: x + 6 =. How is this exaple differet tha the last oe? How would you eed to odify your process to solve it? If studets take exceptio to the - o the right had side, ask the if the radical has bee isolated. Reid the that i the previous siple exaple, the radical was by itself whe we realized the egative o the right had side would give a extraeous solutio. x +. Have you doe this before? Which previously leared processes or rules are you usig? How would you solve: x = x +? What is differet about this exaple tha the last oes? How ca you use the squarig process at the begiig to help you solve? After usig the squarig property you still have a radical i the equatio, ow what do you do? How ay radicals do you have ow? Does it look siilar to the first type of radical equatios you solved? Write the process you would use to cotiue. Ask studets to suarize the process of solvig radical equatios of the types studied. Give studets a worksheet with several radical equatios ( ad radicals, soe with real solutios, ad soe with o real solutio) to coplete. Have the use the process they wrote dow. Is there ore tha oe way to solve? Copare with your classates. Ask studets how they would fid: ( ) Solve higher order radical equatios. Give studets a cube root equatio like x + = ad ask the to thik about how they would solve it. What is differet about this equatio? Ca we square both sides of the equatio like i our previous exaples? Why or why ot? If we squared both sides of the equatio to reove the square root, what do you thik we should do to reove a cube root? What about the th root? Give aother siple cube root equatio, but this tie with a egative uber o the right had side like: x + 6 =. Soe studets should reeber that egatives were a sig of extraeous solutios (i.e. o solutio) with the square root. Ask everyoe to cosider if this will be a issue with cube roots? To break the previous cocept dow, cosider a very siple exaple like x =. Metally, ask studets to fid a uber whose cube root is -. Ca you fid the uber? What is it? Have studets draw the coclusio that egatives are ot a issue with cube roots. With square roots (a eve idex of ) egatives gave a extraeous solutio ad with cube roots (a odd idex of ) egatives produced a real solutio. Ask the studets to give a patter The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

16 for all eve root equatios ad odd root equatios. Give soe practice probles of varyig difficulty (oe root oly, cube roots ad higher). Give at least oe proble with a fractioal power like: ( x 7) 5 8 =. Reid studets to cosider what a fractioal power represets. Quadratic Equatios Overview of Objectives, studets should be able to:. Solve quadratic equatios a. Solve quadratic equatios usig factorig b. Solve quadratic equatios usig the square root property. Solve quadratic equatios usig the quadratic forula. Solve probles usig quadratic equatios. Objectives: Solve quadratic equatios o Solve quadratic equatios usig factorig o Solve quadratic equatios usig the square root property Mai Overarchig Questios:. How do you solve quadratic equatios?. How ay solutios ca we expect to get? Activities ad Questios to ask studets: Give studets the geeral quadratic equatio ax + bx + c = 0 How do you kow a equatio is quadratic? What properties does it have? Give studets a siple product such as 0 ad ask for the result. The reverse the order ad ask for the result of 0. Ask the studets if they otice ay siilarities betwee the two siple expressios. Why is the fial result the sae i each case? What requireet ust be et for the product of two ubers to be 0? Next, give the studets the siple equatio: a b = 0. What are the possible solutios to this equatio? Is there ore tha oe solutio? Have studets establish the zero product property: if the product of two factors is 0, the either The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

17 factor is 0. Give studets a very siple quadratic equatio to solve like ( x + )(x + ) = 0 How would we solve this equatio? If ecessary, reid studets of what has bee discussed so far this discussio. Next give studets a siple quadratic equatio: x + x = 0 Ask studets how they ight attept to solve the equatio. How ca we use the zero product property to aid i solvig the equatio? If we eed a product to use the property, how ca we trasfor our su of ters ito a product? If studets have trouble, give studets a arithetic exaple to illustrate the poit: How would we write 0 as a product? The, have studets solve x + x = 0. What steps did you use to solve the equatio? Give studets aother equatio to solve usig a differet factorig ethod like x + x + = 0 What differeces did you otice i solvig this equatio? Give studets oe additioal equatio that does ot have 0 o the right had side: x + x + =. What additioal steps ight be ecessary? Suarize the process. I each exaple, how ay solutios did we get? Is there a relatioship betwee the degree of the equatio ad the uber of solutios? Square Root Property: Give studets a very siple quadratic equatio to solve like: x =. If studets oly give x = as the solutio ask the if there ay other solutios. If ecessary, ask the how ay solutios we orally have whe solvig quadratic equatio. Next, ask studets (this has already bee discussed) what is. How ca we get rid of a square o a variable? Now, goig back to x = ask the studets what other process we ight use to solve the equatio. If we take the square root of both sides, what additioal steps do we eed to get both solutios? Have studets suarize the process of solvig quadratic equatios by takig square roots. Give studets aother quadratic equatio to solve: x = 8. What is differet about this equatio? What step could we use to ake the equatio look siilar to the first oe? x The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

18 Solve quadratic equatios usig the quadratic forula Have studets suarize the steps to solve this equatio. What steps would be ecessary to solve: 5x + = 7? What is differet about this equatio? What step or steps would be ecessary to ake it look like the first equatio? Fially, have studets suarize the steps to solve ( x + ) = 5. What is differet about the squared portio of this equatio? After takig the square root of both sides what additioal step or steps is ecessary? Give studets the geeral quadratic equatio ad quadratic forula: ax + bx + c = 0 ad b ± b ac x =. a What do you otice about the forula? What does the forula give us? How ay solutios should we get? I what istaces would we get solutio? No solutios? Ask studets how could they idetify what the a, b, ad c values are. Give studets the exaple x + x + = 0. Ask studets to suarize the process they use to solve the equatio. Ask i particular that they suarize the steps i siplifyig the expressio. What about x = x +? What are the a, b, ad c values? What additioal step do we eed before pluggig ito the forula?. Solve probles usig quadratic equatios.. Have studets revisit the projectile proble. Agai, sice we have t itroduced the graph of a quadratic equatio, give the studets a graph of the height of projectile vs. tie.. Although they do t have the tools to calculate the axiu height, ask studets where they thik the axiu height would occur.. Have studets describe at what poit o the graph the ball would hit the groud? What is the height at this poit? 5. Give studets a quadratic equatios that represets the height of the projectile such as h = t + 5t 6. If we are asked to figure out whe the projectile hits the groud, what value should we let h be? How ca we use what we have leared to figure out the issig tie values? 6. Oce studets have solved the equatio, ask the how ay solutios they foud. Are both solutios valid? Why or why ot? Which solutio is valid? The cotets of this website were developed uder Cogressioally-directed grat (P6Z080078) fro the U.S. Departet of Educatio. However, those cotets do ot ecessarily represet the policy of the U.S. Departet of Educatio, ad you should ot assue edorseet by the Federal Goveret.

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

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