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Aim: How can the sum and the product of the roots help in writing a quadratic equation?

Aim: How can the sum and the product of the roots help in writing a quadratic equation?. Do Now:. Write a quadratic equation whose roots are r 1 and r 2. write the roots. set equal to zero. simplify. (r 1 r 2 ) is the product of the roots. (r 1 + r 2 ) is the sum of the roots.

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Aim: How can the sum and the product of the roots help in writing a quadratic equation?

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  1. Aim: How can the sum and the productof the roots help in writing a quadratic equation? Do Now: Write a quadratic equation whose roots are r1 and r2.

  2. write the roots set equal to zero simplify (r1r2) is the product of the roots (r1 + r2) is the sum of the roots General Roots Write a quadratic equation whose roots are r1 and r2. x = r1 x = r2 x – r1 = 0 x – r2 = 0 (x – r1)(x – r2) = 0 multiply binomials x2 – r1x – r2x + r1r2 = 0 x2 – (r1 + r2)x + r1r2 = 0 the b term the c term

  3. multiply by 1/a compare once more x2 – (r1 + r2)x + r1r2 = 0 -(r1 + r2) = b/a r1r2 = c/a a, b, c ––– r1 and r2 the sum and product of roots x2 – (r1 + r2)x + r1r2 = 0 ax2 + bx + c = 0 - standard form for a to equal 1 1/a(ax2 + bx + c = 0) or (r1 + r2) = -(b/a) the sum of the roots = -(b/a) whena = 1 the product of the roots = c/a

  4. Write a quadratic equation whose roots are 1. sum of roots = 2. product of roots = 3. let a = 1 4. substitute a = 1, b = -6, and c = 4 in Using Sum & Product of Roots (r1 + r2) = -(b/a) the sum of the roots = -(b/a) r1r2 = c/a the product of the roots = c/a -(b/a) = 6 c/a = 4 then -(b/a) = 6; b = -6 then c/a = 4; c = 4 check ax2 + bx + c = 0 x2 – 6x + 4 = 0

  5. (r1 + r2) = -(b/a) r1r2 = c/a a. b. Model Problems For the quadratic equation 2x2 + 5x + 8 = 0 find: a. the sum of its roots b. the product of its roots the sum of the roots = -(b/a) the product of the roots = c/a a = 2, b = 5, c = 8 (r1 + r2) = -(b/a) (r1 + r2) = -(5/2) r1r2 = c/a r1r2 = 8/2 = 4

  6. (r1 + r2) = -(b/a) the sum of the roots = -(b/a) r1r2 = c/a the product of the roots = c/a (r1 + r2) = -(b/a) r1r2 = c/a let a = 1 substitute a = 1, b = 0, and c = 25 in Model Problems Write a quadratic equation whose roots are 5i and -5i (5i + -5i) = 0 = -(b/a) (5i)(-5i) = 25 =c/a then -(b/1) = 0; b = 0 then c/1 = 25; c = 25 ax2 + bx + c = 0 x2 + 25 = 0 check

  7. (r1 + r2) = -(b/a) r1r2 = c/a let a = 1 substitute a = 1, b = -6, and c = 13 in Model Problems If one root of a quadratic is 3 + 2i, what is the other root? 3 – 2i Write the quadratic equation having these roots. (3 – 2i) + (3 + 2i) = 6 = -(b/a) (3 – 2i)(3 + 2i) = 13 =c/a then -(b/1) = 6; b = -6 then c/1 = 13; c = 13 ax2 + bx + c = 0 x2 – 6x + 13 = 0 check

  8. substitute 4 for x solve for k factor & solve Model Problems If one root of x2 – 6x + k = 0 is 4, find the other root. Method 1: (4)2 – 6(4) + k = 0 16 – 24 = -k k = 8 x2 – 6x + 8 = 0 (x – 4)(x – 2) = 0 x = 4, x = 2 the other root is 2

  9. (r1 + r2) = -(b/a) r1r2 = c/a let r1 = 4 Model Problems If one root of x2 – 6x + k = 0 is 4, find the other root. Method 2: a = 1, b = -6 (4 + r2) = -(-6/1) 4 + r2 = 6 r2 = 2

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