Sum and Product of Roots

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.

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  =  +  =

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

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

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 =

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 = –1 + 2 + 1 = 2

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