1. Sum and Product of Roots

2. Relations between Roots and Coefficients
Suppose  and  are the roots of ax2 + bx + c = 0.
We can express  +  and  in terms of a, b and c.

3. Compare the coefficient of x and the constant term.
ax2 + bx + c = 0
(x – )(x – ) = 0 2 = + a c x b
x2 – x – x +  = 0
x2 – ( + )x +  = 0
Compare the coefficient of x and the constant term.
Sum of roots =
Product of roots = – a b c
 =
 +  =

4. For each of the following quadratic equations, find the sum and the product of its roots.
Sum of roots =
Product of roots = 2 7 – =
2x2 + 7x = 0
2x2 + 7x + 0 = 0
3x – x2 = 1
Sum of roots =
Product of roots = 3 1 ) ( = –
x2 – 3x + 1 = 0

5. Follow-up question
It is given that the sum of the roots of x2 – (3 – 4k)x – 6k = 0 is –9. (a) Find the value of k. (b) Find the product of the roots.
For the equation x2 – (3 – 4k)x – 6k = 0,
(a)
sum of roots = a b –
Sum of roots =
= 3 – 4k
∴ –9 = 3 – 4k
k = 3

6. Follow-up question
It is given that the sum of the roots of x2 – (3 – 4k)x – 6k = 0 is –9. (a) Find the value of k. (b) Find the product of the roots.
(b)
Product of roots =
= –6(3)
= –18 a c
Product of roots =

7. If a and b are the roots of the quadratic equation x2 – 2x – 1 = 0, find the values of the following expression. (a) (a + 1)(b + 1) (b) a2 + b 2
a + b =
= 2,
ab =
= -1
(a)
(a + 1)(b + 1) = ab + b + a + 1
= ab + (a + b) + 1
= –
= 2

8. If a and b are the roots of the quadratic equation x2 – 2x – 1 = 0, find the values of the following expression. (a) (a + 1)(b + 1) (b) a2 + b 2
a + b =
= 2,
ab =
= -1
(b)
a2 + b 2 = (a2 + 2ab + b 2) – 2ab
= (a + b)2 – 2ab
= (2)2 – (–1)
= 5