1. Given that 4x²- 16x + 15 = a(x-p)²+ q for all values of x. (a) Find the values of the constants a, p and q.

a = 4, p = 2, q = - 1

Step-by-step explanation:

Expand the right side of the identity, then compare the coefficients of like terms with those on the left side.

a(x - p)² + q ← expand (x - p)² using FOIL

= a(x² - 2px + p²) + q  ← distribute parenthesis

= ax² - 2apx + ap² + q

Compare coefficients of x² term

a = 4

Compare coefficients of x- term

- 2ap = - 16, that is

- 2(4)p = - 16

- 8p = - 16 ( divide both sides by - 8 )

p = 2

Compare constant terms

ap² + q = 15 , that is

4(2)² + q = 15

16 + q = 15 ( subtract 16 from both sides )

q = - 1

Thus a = 4, p = 2, q = - 1