Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. 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.

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. simplify set equal to zero write the roots General Roots Write a quadratic equation whose roots are r 1 and r 2. x = r 1 x = r 2 x – r 1 = 0x – r 2 = 0 (x – r 1 )(x – r 2 ) = 0 x 2 – r 1 x – r 2 x + r 1 r 2 = 0 multiply binomials x 2 – (r 1 + r 2 )x + r 1 r 2 = 0 (r 1 + r 2 ) is the sum of the roots (r 1 r 2 ) is the product of the roots the b term the c term

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. multiply by 1/a a, b, c ––– r 1 and r 2 ax 2 + bx + c = 0 - standard form x 2 – (r 1 + r 2 )x + r 1 r 2 = 0 the sum and product of roots 1/a(ax 2 + bx + c = 0) -(r 1 + r 2 ) = b/a or (r 1 + r 2 ) = -(b/a) r 1 r 2 = c/a the sum of the roots = -(b/a) the product of the roots = c/a x 2 – (r 1 + r 2 )x + r 1 r 2 = 0 compare once more for a to equal 1 when a = 1

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. Using Sum & Product of Roots (r 1 + r 2 ) = -(b/a) r 1 r 2 = c/a the sum of the roots = -(b/a) the product of the roots = c/a Write a quadratic equation whose roots are 1. sum of roots = -(b/a) = 6 2. product of roots = c/a = 4 3. let a = 1then -(b/a) = 6; b = -6 then c/a = 4; c = 4 4. substitute a = 1, b = -6, and c = 4 in ax 2 + bx + c = 0x 2 – 6x + 4 = 0 check

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. b. a. Model Problems For the quadratic equation 2x 2 + 5x + 8 = 0 find: a. the sum of its roots b. the product of its roots (r 1 + r 2 ) = -(b/a) r 1 r 2 = c/a the sum of the roots = -(b/a) the product of the roots = c/a a = 2, b = 5, c = 8 (r 1 + r 2 ) = -(b/a)(r 1 + r 2 ) = -(5/2) r 1 r 2 = c/ar 1 r 2 = 8/2 = 4

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. Model Problems Write a quadratic equation whose roots are 5i and -5i (r 1 + r 2 ) = -(b/a) r 1 r 2 = c/a the sum of the roots = -(b/a) the product of the roots = c/a (r 1 + r 2 ) = -(b/a) (5i + -5i) = 0 = -(b/a) r 1 r 2 = c/a (5i)(-5i ) = 25 = c/a let a = 1then -(b/1) = 0; b = 0 then c/1 = 25; c = 25 substitute a = 1, b = 0, and c = 25 in ax 2 + bx + c = 0x = 0 check

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. 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. (r 1 + r 2 ) = -(b/a) (3 – 2i) + (3 + 2i) = 6 = -(b/a) r 1 r 2 = c/a (3 – 2i)(3 + 2i) = 13 = c/a let a = 1then -(b/1) = 6; b = -6 then c/1 = 13; c = 13 substitute a = 1, b = -6, and c = 13 in ax 2 + bx + c = 0x 2 – 6x + 13 = 0 check

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. Model Problems If one root of x 2 – 6x + k = 0 is 4, find the other root. Method 1: substitute 4 for x(4) 2 – 6(4) + k = 0 solve for k 16 – 24 = -k k = 8 x 2 – 6x + 8 = 0 factor & solve (x – 4)(x – 2) = 0 x = 4, x = 2 the other root is 2

Aim: Sum & Product of Roots Course: Adv. Alg. & Trig. let r 1 = 4 Model Problems If one root of x 2 – 6x + k = 0 is 4, find the other root. Method 2: a = 1, b = -6 (r 1 + r 2 ) = -(b/a) r 1 r 2 = c/a (4 + r 2 ) = -(-6/1) 4 + r 2 = 6 r 2 = 2