Math 135 Circles and Completing the Square Examples


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1 Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = We wish to hve method for finding b when is n expression. For instnce, you should remember tht 2 + b + b 2 is perfect squre, becuse it is exctly + b) 2. How would you turn the expression x 2 + x into perfect squre? A moment of thought should convince you tht if we dd 2 )2 to x 2 + x we obtin perfect squre, becuse x + 2 )2 = x 2 + x + 2 )2. The ddition of 2 )2 is clled completing the squre, becuse the new expression cn now be written s squre of some other expression. Exmple 1. Complete the squre: x 2 + 4x = 0 x 2 + 4x = 0 x 2 + 4x + 4) = 4 x + 2) 2 = 4 We hve dded the squre of hlf the coefficient of x to the originl eqution, nd therefore to mintin equlity it ws necessry to dd the sme mount to the other side of the eqution. Wrning 2. The coefficient of x 2 must be equl to 1 in order to complete the squre. Exmple 3. Complete the squre: 2x 2 + 8x = 0 2x 2 + 8x = 0 2x 2 + 4x) = 0 2x 2 + 4x + 4) = 8 2x + 2) 2 = 8 We dded 4, the squre of hlf the coefficient of x, inside the prentheses. Note tht this mounts to dding 8 to the left side of the eqution, becuse everything inside the prentheses is multiplied by 2. Therefore, to mintin equlity we dd 8 to the right side of the eqution. In cse we cnnot set our expression equl to 0, we must subtrct whtever number we dd to the expression: Exmple 4. Complete the squre: 2x 2 + 8x 2x 2 + 8x = 2x 2 + 4x) = 2x 2 + 4x + 4) 8 = 2x + 2) 2 8 Exmple 5. x h) 2 + y k) 2 = r 2 is the eqution of circle of rdius r centered t the point h, k). Using the method of completing the squre twice) find the rdius nd center of the circle given by the eqution x 2 + y 2 + 8x 6y + 21 = 0. x 2 + y 2 + 8x 6y + 21 = 0 1) x 2 + 8x) + y 2 6y) = 21 2) x 2 + 8x + 16) + y 2 6y + 9) = ) x + 4) 2 + y 3) 2 = 4 4) We hve now the form x 4)) 2 + y 3) 2 = 2 2 which is circle of rdius r = 2 centered t the point h, k) = 4, 3). University of Hwi i t Māno 52 R Spring
2 Mth 135 Circles nd Completing the Squre Exmples Deriving the Qudrtic Formul Given qudrtic eqution, i.e. n eqution of this form: x 2 + bx + c = 0, 0 5) where, b, nd c re rel numbers, we wish to hve formul tht will give us the explicit vlues of x for which the qudrtic eqution is zero. Tht is, we need formul tht produces x 1 nd x 2 such tht x bx 2 + c = 0 nd x bx 2 + c = 0 6) The qudrtic formul tells us exctly how to find our set of solutions {x 1, x 2 }, but it lso tells how lrge this set is. We cn hve two distinct solutions nd this hppens whenever the discriminnt is positive number. We cn hve just one solution if the discriminnt is zero. In this cse we sy tht the root x 1 = x 2 ) hs multiplicity 2, becuse it occurs twice. Finlly, when the discriminnt is negtive number, we hve squre root of negtive number nd hence no rel) solutions. Recll the qudrtic formul: x = b ± b 2 4c where the discriminnt is equl to b 2 4c 7) How do we know tht this is indeed correct? We cn pply the method of completing the squre to our qudrtic eqution 1) nd verify tht eqution 2) is correct. Here re the detils: x 2 + bx + c = 0 8) x 2 + b x + c = 0 9) x 2 + b x = c 10) ) x 2 + b 2 ) 2 b b x + = c 2 2 x 2 + b ) 2 ) 2 b b x + = c ) 2 4 c 2 ) 2 c ) 12) 13) 14) ) 2 4 4c 15) 2 ) 2 = 16) = ± 17) = ± 18) 4 2 = ± 19) University of Hwi i t Māno 53 R Spring
3 Mth 135 Circles nd Completing the Squre Exmples x = b + ± x = b ± b 2 4c So which of the bove steps do we cll completing the squre? The nswer is 4) to 7); the rest del with writing the eqution in the form x = something. Let s review: Suppose you re given your fvorite qudrtic x 2 + bx + c nd need to solve for x. You re no longer mused by fctoring nd decide to complete the squre insted. Step 1: Check the coefficients. If = 0 you don t need to complete the squre. If 1 then you need to fctor out. So suppose tht 1 nd 0. [ x 2 + bx + c = x 2 + b x + c ] 22) Step 2: Group the x terms together. You complete the squre only on the terms contining the vrible x. Notice tht inside the brckets [ ] we now hve new qudrtic eqution with coefficients = 1, b = b nd c = c. x 2 + bx + c = [x 2 + b ) x + c ] Step 3: Complete the squre: dd the squre of hlf of the coefficient of x to the terms in side the prentheses ). [ x 2 + bx + c = x 2 + b ) ) ] 2 b x + + c 24) Step 4: Up until now we hve not ltered the eqution, but dding something to the right side requires subtrcting the sme number. We hve dded b2 inside the brckets [ ] nd everything inside [ ] is multiplies by. Therefore, to keep the eqution unchnged, we now subtrct from the right side the number b2 nd obtin [ x 2 + bx + c = x 2 + b ) ) ] 2 b x + + c b2 25) 4 20) 21) 23) Step 5: Simplify. The term in the prentheses ) is perfect squre nd so [ x 2 + bx + c = ) ] 2 + c b2 4 = ) ) This form should look fmilir. If we were to set line 22) equl to zero we would hve the stndrd qudrtic eqution. Then dividing by legl since 0) nd moving terms round returns us to eqution 12). University of Hwi i t Māno 54 R Spring
4 Mth 135 Circles nd Completing the Squre Exmples Viete s Equtions, or how to pick out the correct pir of solutions to qudrtic eqution... Proposition 6. Given qudrtic eqution with rel coefficients, b, c x 2 + bx + c = 0, 0 If the solutions exist, then they hve the following form x 1 = b + b 2 4c x 2 = b b 2 4c nd they obey the following lgebric equtions: x 1 + x 2 = b x 1 x 2 = c If you re given qudrtic eqution to solve nd re llowed to use the qudrtic formul, then you my follow these steps nd sve yourself some work. Step 1: Mke sure tht the solutions exist, i.e. b 2 4c 0 Step 2: Look t the qudrtic eqution you hve to solve nd determine the vlues of, b, c nd compute b nd c. Step 3: Compute x 1 +x 2 nd x 1 x 2 for ech set of solutions your re given s choice. Step 4: Compre the results of steps 2 nd 3. If you find mtch, you hve found the solution. If there is no mtch, then none of the possible choices is solution. Is it possible to hve more thn one set of mtching solutions?) University of Hwi i t Māno 55 R Spring
5 Mth 135 Circles nd Completing the Squre Worksheet Using the method of completing the squre, put ech circle into the form. x h) 2 + y k) 2 = r 2 Then determine the center nd rdius of ech circle. 1. x 2 + y 2 10x + 2y + 17 = x 2 + y 2 + 8x 6y + 16 = x x + 9y 2 18y + 64 = x 2 4x + 4y 2 59 = 0. y h, k + r) h r, k) Center t h, k) h + r, k) Rdius r x h, k r) Smple Midterm 3 A B C D 8 A B C D 11 A B C D 29 A B C D 31 A B C D 32 A B C D 35 A B C D Smple Finl University of Hwi i t Māno 56 R Spring
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