Find the values of a and b such that

x^2 + 2x - 7 = (x+a)^2 + b

The values of a and b are 1 and -8

Step-by-step explanation:

Let us solve the question by comparing the two sides.

∵ x² + 2x - 7 = (x + a)² + b

→ Let us solve the bracket on the right side

∵ (x + a)² = (x)(x) + 2(x)(a) + (a)(a)

∴ (x + a)² = x² + 2ax + a²

→ Substitute it in the right side above

∴ x² + 2x - 7 = x² + 2ax + a² + b

→ Compare the like terms on both sides (terms of x², terms of x

   and numerical terms)

∵ The terms of x are 2x and 2ax

→ Equate them

∵ 2x = 2ax

→ Divide both sides by 2x

∴ =

∴ 1 = a

∴ The value of a = 1

∵ The numerical terms are -7 and a² + b

→ Equate them

∵ -7 = a² + b

→ Substitute a by 1

∴ -7 = (1)² + b

∴ -7 = 1 + b

→ Subtract 1 from both sides

∵ -7 - 1 = 1 - 1 + b

∴ -8 = b

∴ The value of b = -8

∴ The values of a and b are 1 and -8