Question

Express (x^2 + 4x - 7) in the form ((x + a)^2 - b) where (a) and (b) are integers. Please show workings.

a) (x^2 + 4x + 9)
b) ((x + 2)^2 - 16)
c) ((x + 2)^2 - 9)
d) (x^2 + 4x + 7)

Answer 1

To express (x^2 + 4x - 7) in the form ((x + a)^2 - b), complete the square by adding and subtracting (4/2)^2, resulting in the expression ((x + 2)^2 - 11). Option c) is close but has the incorrect constant.

Step-by-step explanation:

To express (x^2 + 4x - 7) in the form ((x + a)^2 - b), where (a) and (b) are integers, we will complete the square for the quadratic expression.

Here are the steps:

1. Take the coefficient of x, which is 4, divide it by 2, and square it to get the constant term that will complete the square.
   (4/2)^2 = 2^2 = 4.
2. Add and subtract this value inside the parenthesis to maintain the equality:
   x^2 + 4x + 4 - 4 - 7.
3. Now, group the perfect square trinomial and the constants separately:
   ((x + 2)^2 - 4 - 7).
4. Combine the constants:
   ((x + 2)^2 - 11).

This results in the expression in the desired form, and by comparing it with the options given, option c) ((x + 2)^2 - 9) is almost correct, however, the correct constant should be -11 instead of -9 to match the expression we derived.