Quadratics  Build Quadratics From Roots


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1 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 factoring 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. The solutions are and Set each solution equal to x x = or x = Make each equation equal zero + + Subtractfrom first, add to second x =0 or x + =0 These expressions are the factors (x )(x + )=0 FOIL x +x x 8 Combine like terms x x 8=0 If one or both of the solutions are fractions we will clear the fractions by multiplying by the denominators. Example. The solution are and Set each solution equal tox x = or x = Clear fractions by multiplying by denominators x = or x = Make each equation equal zero Subtract from the first, subtractfrom the second x =0 or x =0 These expressions are the factors (x )(x ) =0 FOIL 1x 9x 8x + 6=0 Combine like terms 1x 17x + 6=0 1
2 If 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. The solutions are and x = ± x = x =0 Write as one expression equal tox Subtractfrom both sides 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) = a + ab +b Example. The solutions are 5 and +5 x =±5 x =±5 x x+=5 x x +=50 Write as one expression equal tox Isolate the square root term Subtract from both sides Subtract 50 x x 6 = 0 World View Note: Before the quadratic formula, before completing the square, before factoring, quadratics were solved geometrically by the Greeks as early as 00 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. The solutions are + and x = ± x = ± Write as one expresion equal tox Clear fraction by multiplying by Isolate the square root term
3 x =± 16x 16x + = 16x 16x + 1=0 Subtractfrom both sides Subtract The process used for complex solutions is identical to the process used for radicals. Example 6. The solutions are 5i and +5i Write as one expression equal tox x =±5i Isolate the i term Subtract from both sides x =±5i x 8x + 16 = 5i i = 1 x 8x + 16 = Add 5 to both sides x 8x + 1 =0 Example 7. The solutions are 5i and +5i x = ± 5i x =±5i x =±5i x 1x + 9=5i i = 1 x 1x +9= 5 Write as one expression equal to x Clear fraction by multiplying by denominator Isolate the i term Subtract from both sides Add 5 to both sides x 1x + =0 Attribution.0 Unported License. (
4 9.5 Practice  Build Quadratics from Roots From each problem, find a quadratic equation with those numbers as its solutions. 1), 5 ) 0, 5), 7) 0, 0 9), 11 11), 1 1) 1, 1 15) 7, 17) 1, ) 6, 1 9 1) ± 5 ) ± 1 5 5) ± 11 7) ± 9) ±i 1 1) ± 6 ) 1 ± i 5) 6 ±i 7) 1 ± 6 9) 6 ± i 8 ), 6 ) 1, 1 6) 0, 9 8), 5 10), 1 1) 5 8, 5 7 1) 1, 16), 9 18) 5, 1 0) 5,0 ) ± 1 ) ± 7 6) ± 8) ± 11i 0) ± 5i ) ± ) ± i 6) 9 ±i 5 8) ± 5i 0) ± i 15 Attribution.0 Unported License. (
5 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) x 7x + 10=0 ) x 9x + 18 =0 ) x x + 0 =0 ) x 1x + 1 = 0 5) x 8x + 16 =0 6) x 9x = 0 7) x = 0 8) x + 7x + 10 = 0 9) x 7x =0 10) x x =0 11) 16x 16x +=0 1) 56x 75x + 5 =0 1) 6x 5x + 1=0 1) 6x 7x + =0 15) 7x 1x + 1 = 0 16) 9x 0x +=0 17) 18x 9x 5=0 18) 6x 7x 5=0 19) 9x + 5x 6=0 0) 5x +x = 0 1) x 5 = 0 ) x 1=0 ) 5x 1=0 ) x 7=0 5) x 11 = 0 6) x 1 = 0 7) 16x =0 8) x + 11 = 0 9) x + 1 = 0 0) x + 50 = 0 1) x x =0 ) x + 6x +7=0 ) x x + 10 = 0 ) x + x + 0 = 0 5) x 1x + 9 = 0 6) x + 18x + 86 = 0 7) x + x 5=0 8) 9x 1x + 9 =0 9) 6x 96x + 8 =0 0) x + 8x + 19 = 0 Attribution.0 Unported License. ( 5
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