1. Sum and Product of Roots
Quadratic Functions
Sum and Product of Roots

2. Purpose
Find sum/product of roots without knowing the actual value of x.
Use the sum/product of roots to solve for other results

3. Derivation of Formula
Given α and β are roots of a quadratic equation, we can infer that
x1 = α and x2 = β , where α and β are just some numbers.
For example, x2 – 5x + 6 = 0 has roots α and β  means that α = 2 and β = 3

4. However…
Sometimes, we do not need the values of x to help us solve the problem.
Knowing the relationship between the quadratic equation and its roots helps us save time

5. Derivation of Formula This tells us that: (x – α)(x – β) = 0
ax2 + bx + c = 0 has roots α and β –(1)
This tells us that:
(x – α)(x – β) = 0
Expanding, we have:
x2 – αx – βx + αβ = 0
x2 – (α + β)x + αβ = 0
So, if we compare the above with (1):
x x + = 0
Result: αβ = , α + β =

6. In other words… x2 – (α + β)x + αβ = 0
ax2 + bx + c = 0 has roots α and β
x2 – (α + β)x + αβ = 0
x2 – (sum of roots)x + product of roots = 0

7. Application – Example 1 (TB Pg 55)
2x2 + 6x – 3 = 0 has roots α and β, find the value of
(a)
Without finding α and β, we know the value for α + β and αβ
α + β = – 3 and αβ = – 3/2
Hence,

8. Application – Example 1 (TB Pg 55)
2x2 + 6x – 3 = 0 has roots α and β, find the value of
(b)
We already know that: α + β = – 3 and αβ = – 3/2
Expanding (b), we have

9. Application – Example 2 (TB Pg 55)
x2 – 2x – 4 = 0 has roots α and β, find the value of
(a)
Step 1: α + β = 2 and αβ = – 4
To find (a), we use the algebraic property
Rearranging:

10. Application – Example 2 (TB Pg 55)
x2 – 2x – 4 = 0 has roots α and β, find the value of
(b)
Step 1: α + β = 2 and αβ = – 4
To find (b), we use the algebraic property
Rearranging:
Hence, α − β =

11. Application – Example 2 (TB Pg 55)
x2 – 2x – 4 = 0 has roots α and β, find the value of
(c)
Step 1: α + β = 2 and αβ = – 4
To find (a), we use the algebraic property
Rearranging:

12. Application – Example 3 (TB Pg 56)
2x2 + 4x – 1 = 0 has roots α and β, form the equation with roots α2 and β2
Remember!
x2 – (sum of roots)x + product of roots = 0
x2 – (α2 + β2)x + α2β2 = 0
So in order to get the equation, I must find α2 + β2 and α2β2
How?