Student Academic Learning Services Page 1 of 7


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1 Student Academic Learning Services Page 1 of 7 Quadratic Equations What is a Quadratic Equation? A quadratic equation is an equation where the largest power on a variale is two. All quadratic equations can e written in the form: Where a,, and c can e any real numer, including zero. The eamples elow are all quadratic equations. Below each equation the values of a,, and c are shown. Graphs of Quadratic Equations When graphing a quadratic equation it is much easier to put it into the following form: When graphed, a quadratic function will look like a curve in the shape of a U or an upsidedown U. A few eamples of quadratic functions that have een graphed are shown elow. When y is set to zero, equations like the eamples aove are formed. Therefore, solutions to these equations can e found y determining where the function crosses the ais. These graphs suggest that a quadratic equation may have two, one, or zero solutions depending on how it intersects the ais. Graph of y = Graph of y = Has intercepts at =  and 1 Has no intercepts Student Services Building (SSB), Room et. 491 This document last updated: 1//010
2 Student Academic Learning Services Page of 7 Factoring Quadratics There are several different techniques for factoring quadratic equations, and they all apply to different situations. It is important to e ale to factor quadratics fully, as we will see when we try to solve them without using graphs. First, for a quadratic with c = 0, or no constant term, such as eample 1 elow, you can simply factor the out normally and you re done. Second, for a quadratic with = 0, or no term, such as eample elow, you can use the difference of squares method. This method will only work out evenly if a is a perfect square, and is the negative of a perfect square. The middle eample has een factored using this method. To check that it is factored correctly, try epanding the result. Third, for a quadratic with 0, and c 0, such as eample 3 elow, you can use a method that is the reverse of the FOIL method for multiplying two inomials. To check that it is factored correctly, try epanding the result. Eample 1 Eample Eample 3 + = (+) 4 9 = (+3)(3) = (+)(+1) Difference of Squares The difference of squares method for factoring quadratics is simple. It makes use of the following properties: This property can e used to factor the types of quadratic equations that have = 0 or the form of the eamples elow. This method works est if a is a perfect square and c is the negative of a perfect square ut can actually e used on any quadratic as long as = 0, and c < 0 (i.e. c is negative). Below are 4 eamples of quadratic equations factored y the use of the difference of squares method. 9 = (3)(+3) 4 49 = (7)(3+7) 18 0 = (9 ) = (3)(3+) = ( )(+ ) Student Services Building (SSB), Room et. 491 This document last updated: 1//010
3 Student Academic Learning Services Page 3 of 7 The Reverse of FOIL For this net method, you must recall the method for multiplying inomials, known as FOIL (First, Outside, Inside, Last). When you multiply two inomials the usual result is a trinomial, which is often a quadratic. When you have a quadratic, it is often possile to use the reverse process to factor it into two inomials. It is often not easy to do so, ut with practice, you get etter at it. Have a look at the eamples elow, and check to see that the result will epand, y using FOIL, ack to the original quadratic = (+3)(+1) (+)(+3) = =(First+Outside+Inside+Last) = = (+3)() A Stepystep Approach Learning how to reversefoil a quadratic isn t easy. The est way to think of it is to go ack, and look at the process of FOIL in reverse. Consider the eamples aove, we have two inomials that, when multiplied together, yield a quadratic. But what is the est way to get those two inomials? Before we delve into solving quadratic equations, there is one other thing you should learn, and that is the discriminant. Not all quadratics are factorale, and the discriminant is very useful ecause it tells us if a quadratic is actually factorale, or if trying to factor it is a waste of time. The Discriminant The discriminant is a tool you can use to see for sure whether a quadratic is factorale. With a,, and c as the previously defined coefficients of the quadratic, the discriminant is calculated as: If the discriminant is less than zero, then the quadratic is not factorale. If the discriminant is greater than, or equal to zero then the quadratic is factorale. And it s just that simple! So use the discriminant whenever you aren t sure if a quadratic is factorale. Otherwise, you may e wasting your time. Student Services Building (SSB), Room et. 491 This document last updated: 1//010
4 Student Academic Learning Services Page 4 of 7 Procedure for Factoring Method 1  When a = 1: Use the following epression as an eample for method 1: Step 1: Write every possile factorization of the numer c. Factors for () (, 1), (, 1), (, 3), (, 3) Step : Find a pair of factors that add up to the numer. Need a pair that adds up to 11 = = = = 1 BINGO!! Step 3: The factorization of the quadratic will look like (+_ )(+_ ), where the two factors (3 and ) will e in the lank spaces. The final solution will then e that Step 4: Check your answer y epanding it out y FOIL. This method works ecause, if you FOIL the result, you end up multiplying the two numers (in this case 3 and ) to get c (), and adding them to get (1). Note though, that if c is negative, then one of the factors will have to e negative, and the other positive. Also, if c is positive, then oth of the two factors will either e negative, or oth of them will e positive. Method  When a is a prime numer: Use the following epression as an eample for method : Step 1: Write every possile factorization of the numer c. Factors for 4 (4, 1), (1, 4), (, ), (, ) Step : Find a pair of factors of c that will add up to the numer AFTER one of the factors has een multiplied y a. Need to find a pair that adds up to 7 when one of them is multiplied y 3. 3(4) + 1 = (1) = 7 BINGO!!! Step 3: The factorization of the quadratic will e (a+_)(+_), where the two factors will e in the lank spaces. The final solution will then e that Student Services Building (SSB), Room et. 491 This document last updated: 1//010
5 Student Academic Learning Services Page of 7 Step 4: Check your answer y epanding it out y FOIL. The only difference here is in Step, where now you have to worry aout the numer in front on the squared. Because that numer is prime, you know that the factorization will look something like this: (a + _)( + _), so that the First operation in FOIL results in the right coefficient for squared. As a result, the a value gets multiplied into one of the two factors of c efore the factors are added together. Method 3  When a is not a prime numer: Use the following equation as an eample of method 3: Step 1: Check to see if you can factor a whole numer from the entire equation to make it a little it easier. If it does factor, forget aout the common factor until the end of the question, and just worry aout the quadratic inside the rackets ( ) Step : Write every possile factorization for the numer a and c. Factors for 8 (8,1), (1,8), (,4) (4,) Factors for 9 (9,1), (1,9), (3,3) Step 3: Find the factors of c that when multiplied y some factors of a will add up to Need to find the pair of factors of 9 that, when multiplied y a pair of factors of 8, will add up to 7. (8)9 + (1)1 = 73 (8)1 + (1)9 = 17 (8)3 + (1)3 = 7 BINGO!! Step 4: The factorization of the quadratic will look similar to that of (a + _ )( +_ ), where the factors of c will e the lank spots and a and will e the factors of a from the question. If the question had a common factor from the start, it is time to ring that ack as part of the question. The final solution will then e that Step : Check your answer y epanding with FOIL. As you can proaly see, this method involves the most work, ecause of all the etra factor pairs you must check in step 3. In this case, we were lucky, and the third try produced the desired result, where as there were 9 other cominations we could have tried. Student Services Building (SSB), Room et. 491 This document last updated: 1//010
6 Student Academic Learning Services Page of 7 Solving Quadratic Equations To solve a quadratic equation of the form a + + c = 0 all you have to do is factor it, and then solve using each individual factor. This usually results in two different solutions. In the net eample we have one of our previously factored quadratics set equal to zero. By factoring the quadratic like this, we can now see two solutions to the equation. After all, if either of (  ) or ( + 3) is equal to zero then the whole left side will e equal to zero. So, we can find two different solutions to this equation y solving the two linear equations  = 0 and + 3 = 0. We end up with the solutions of =, and = 3, and that is what we were looking for! No factors! Check the discriminant! The Quadratic Formula The quadratic formula is a ruteforce formula for solving any quadratic. When there is no factorization to e found for a quadratic, the quadratic formula is a very useful tool. It is etremely useful if the values of a,, and c are very large giving them many more factors. The quadratic formula is fairly straightforward. The one thing that you have to rememer is that, ecause of the +/ sign, you will usually end up with two solutions: one for the + and one for the . The Formula is shown at the top of the net page. Look at the part in the square root. It should look very familiar. That is the discriminant that we discussed aove! It can now e seen that if the discriminant is a negative numer, we will have to take the square root of a negative numer. That is why no solutions eist if the discriminant is less than 0. Student Services Building (SSB), Room et. 491 This document last updated: 1//010
7 Student Academic Learning Services Page 7 of 7 Eamples of Using Quadratic Formula It is est to look at eamples to see how this formula works. There is a lot of work involved when using the quadratic formula, ut the good part is that there will always e a solution (as long as the discriminant is greater than 0) no matter how ugly it looks = 0 In this case, a=1, =, and c=4 3 7 = 0 In this case, a=3, =, and c=7 a (1) 9 3, 8, 1, 4 4ac 4(1)(4) 1 3 ( a ) (3) 84 4ac 4(3)( , 1.44,.7, ) In this step, we split into two separate solutions, one using the plus, and one using the minus. Student Services Building (SSB), Room et. 491 This document last updated: 1//010
Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
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