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

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.

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 r1 and r2.

x = r1 x = r2

x – r1 = 0 x – r2 = 0

(x – r1)(x – r2) = 0

x2 – r1x – r2x + r1r2 = 0

multiply binomials

x2 – (r1 + r2)x + r1r2 = 0

(r1 + r2) is the sumof the roots

(r1r2) is the productof the roots

the b term the c term

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

multiply by 1/a

a, b, c ––– r1 and r2

ax2 + bx + c = 0 - standard form

x2 – (r1 + r2)x + r1r2 = 0the sum and product of roots

1/a(ax2 + bx + c = 0)

x2 b

ax

c

a0

-(r1 + r2) = b/a or (r1 + r2) = -(b/a)

r1r2 = c/a

the sum of the roots = -(b/a)

the product of the roots = c/a

x2 – (r1 + r2)x + r1r2 = 0compare once more

for a to equal 1

when a = 1

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

Using Sum & Product of Roots

(r1 + r2) = -(b/a)

r1r2 = c/a

the sum of the roots = -(b/a)

the product of the roots = c/a

Write a quadratic equation whose rootsare

3 5 and 3 5

1. sum of roots =

3 5 3 5 6 -(b/a) = 6

2. product of roots =

3 5 3 5 4c/a = 4

3. let a = 1 then -(b/a) = 6; b = -6

then c/a = 4; c = 44. substitute a = 1, b = -6, and c = 4 in

ax2 + bx + c = 0 x2 – 6x + 4 = 0

check

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

b.

a.

Model Problems

For the quadratic equation 2x2 + 5x + 8 = 0find: a. the sum of its roots

b. the product of its roots

(r1 + r2) = -(b/a)

r1r2 = c/a

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

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

Model Problems

Write a quadratic equation whose roots are5i and -5i

(r1 + r2) = -(b/a)

r1r2 = c/a

the sum of the roots = -(b/a)

the product of the roots = c/a

(r1 + r2) = -(b/a) (5i + -5i) = 0 = -(b/a)

r1r2 = c/a (5i)(-5i ) = 25 = c/a

let a = 1 then -(b/1) = 0; b = 0

then c/1 = 25; c = 25

substitute a = 1, b = 0, and c = 25 in

ax2 + bx + c = 0 x2 + 25 = 0

check

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

Model Problems

If one root of a quadratic is 3 + 2i, whatis the other root?

3 – 2i

x b b2 4ac

2aWrite the quadratic equation having these roots.

(r1 + r2) = -(b/a) (3 – 2i) + (3 + 2i) = 6 = -(b/a)

r1r2 = c/a (3 – 2i)(3 + 2i) = 13 = c/a

let a = 1 then -(b/1) = 6; b = -6

then c/1 = 13; c = 13

substitute a = 1, b = -6, and c = 13 in

ax2 + bx + c = 0 x2 – 6x + 13 = 0

check

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

Model Problems

If one root of x2 – 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

x2 – 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 r1 = 4

Model Problems

If one root of x2 – 6x + k = 0 is 4, find the other root.

Method 2: a = 1, b = -6

(r1 + r2) = -(b/a)

r1r2 = c/a

(4 + r2) = -(-6/1)

4 + r2 = 6

r2 = 2