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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