**9.5**Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions to quadratics by a method such as factoring or completing the square. Here we will take our solutions and work backwards to find what quadratic goes with the solutions. We will start with rational solutions. If we have rational solutions we can use fac- toring in reverse, we will set each solution equal to x and then make the equation equal to zero by adding or subtracting. Once we have done this our expressions will become the factors of the quadratic. Example 1. Thesolutionsare4and−2 x=4 or x=−2 −4−4 x−4=0 or x+2=0 (x−4)(x+2)=0 x2+2x−4x−8 x2−2x−8=0 Seteachsolutionequaltox Makeeachequationequalzero Subtract4fromfirst,add2tosecond Theseexpressionsarethefactors FOIL Combineliketerms OurSolution +2 +2 If one or both of the solutions are fractions we will clear the fractions by multi- plying by the denominators. Example 2. Thesolutionare2 3and3 or x=3 Seteachsolutionequaltox 4 x=2 Clearfractionsbymultiplyingbydenominators 3 4 3x=2 or 4x=3 −2−2 3x−2=0 or 4x−3=0 (3x−2)(4x−3)=0 12x2−9x−8x+6=0 12x2−17x+6=0 Makeeachequationequalzero Subtract2fromthefirst,subtract3fromthesecond Theseexpressionsarethefactors FOIL Combineliketerms OurSolution −3−3 1If the solutions have radicals (or complex numbers) then we cannot use reverse factoring. In these cases we will use reverse completing the square. When there are radicals the solutions will always come in pairs, one with a plus, one with a minus, that can be combined into “one” solution using ± . We will then set this solution equal to x and square both sides. This will clear the radical from our problem. Example 3. √ √ √ Writeas′′one′′expressionequaltox Squarebothsides Makeequaltozero Subtract3frombothsides OurSolution 3 3 3 Thesolutionsare and− x=± x2=3 −3−3 x2−3=0 We may have to isolate the term with the square root (with plus or minus) by adding or subtracting. With these problems, remember to square a binomial we use the formula (a+b)2=a2+2ab+b2 Example 4. √ √ √ Writeas′′one′′expressionequaltox Isolatethesquarerootterm Subtract2frombothsides Squarebothsides Thesolutionsare2−5 2 and2+5 2 x=2±5 2 −2−2 x−2=±5 2 x2−4x+4=25·2 x2−4x+4=50 −50−50 x2−4x−46=0 √ Makeequaltozero Subtract50 OurSolution World View Note: Before the quadratic formula, before completing the square, before factoring, quadratics were solved geometrically by the Greeks as early as 300 BC! In 1079 Omar Khayyam, a Persian mathematician solved cubic equations geometrically! If the solution is a fraction we will clear it just as before by multiplying by the denominator. Example 5. √ 4 √ 4 √ 4 √ Thesolutionsare2+ 3 and2− 3 Writeas′′one′′expresionequaltox x=2± 3 Clearfractionbymultiplyingby4 4x=2± 3 Isolatethesquarerootterm 2

−2−2 4x−2=± 16x2−16x+4=3 Subtract2frombothsides Squarebothsides Makeequaltozero Subtract3 OurSolution √ 3 −3−3 16x2−16x+1=0 The process used for complex solutions is identical to the process used for radi- cals. Example 6. Writeas′′one′′expressionequaltox Isolatetheiterm Subtract4frombothsides Square bothsides i2=−1 Makeequaltozero Add25tobothsides OurSolution Thesolutionsare4−5iand4+5i x=4±5i −4−4 x−4=±5i x2−8x+16=25i2 x2−8x+16=−25 +25 x2−8x+41=0 +25 Example 7. Thesolutionsare3−5i and3+5i Writeas′′one′′expressionequaltox 2 2 x=3±5i Clearfractionbymultiplyingbydenominator 2 2x=3±5i −3−3 2x−3=±5i 4x2−12x+9=5i2 4x2−12x+9=−25 +25 4x2−12x+34=0 Isolatetheiterm Subtract3frombothsides Squarebothsides i2=−1 Makeequaltozero Add25tobothsides OurSolution +25 Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. (http://creativecommons.org/licenses/by/3.0/) 3

9.5 Practice - Build Quadratics from Roots From each problem, find a quadratic equation with those numbers as its solutions. 1) 2, 5 2) 3, 6 3) 20, 2 4) 13, 1 5) 4, 4 6) 0, 9 7) 0, 0 8) −2,−5 10) 3, −1 12) 9) −4,11 11) 3 4,1 5 8,5 4 7 1 2,1 13) 1 2,2 14) 3 3 3 7,4 15) 16) 2,2 9 17) −1 19) −6,1 21) ±5 23) ±1 3,5 5 3,−1 18) 6 2 20) −2 22) ±1 24) ± 26) ±2 3 28) ±11i 30) ±5i 2 32) −3± 34) −2±4i 36) −9±i 5 38) 3 5,0 9 √ 7 5 √ √ 25) ± 11 √ 3 27) ± 4 √ √ √ 29) ±i 13 31) 2± 33) 1±3i 35) 6±i 3 √ 2 6 √ √ √ 2±5i −1± 6 37) 2 √ √ 6±i 2 8 −2±i 15 2 39) 40) Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. (http://creativecommons.org/licenses/by/3.0/) 4

9.5 Answers - Build Quadratics from Roots NOTE: There are multiple answers for each problem. Try checking your answers because your answer may also be correct. 1) x2−7x+10=0 2) x2−9x+18=0 3) x2−22x+40=0 4) x2−14x+13 = 0 5) x2−8x+16=0 6) x2−9x = 0 7) x2=0 15) 7x2−31x+12=0 16) 9x2−20x+4=0 17) 18x2−9x−5=0 18) 6x2−7x−5=0 19) 9x2+53x−6=0 20) 5x2+2x=0 29) x2+13=0 30) x2+50=0 31) x2−4x−2=0 32) x2+6x+7=0 33) x2−2x+10=0 34) x2+4x+20=0 21) x2−25=0 22) x2−1=0 23) 25x2−1=0 24) x2−7=0 25) x2−11=0 26) x2−12=0 27) 16x2−3=0 28) x2+121=0 8) x2+7x+10=0 35) x2−12x+39=0 36) x2+18x+86=0 9) x2−7x−44=0 10) x2−2x−3=0 11) 16x2−16x+3=0 12) 56x2−75x+25=0 13) 6x2−5x+1=0 14) 6x2−7x+2=0 37) 4x2+4x−5=0 38) 9x2−12x+29=0 39) 64x2−96x+38=0 40) 4x2+8x+19=0 Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. (http://creativecommons.org/licenses/by/3.0/) 5