Decide whether each of the following pairs of expressions are equivalent for all values of x.
If they are equivalent, show how you can be sure. If
they are not, justify your reasoning completely.
Pair 1: (x + 2)2 and x² + 4

Pair 2: (x + 5)2 and x² + 10x + 25
*are these pairs equal?*

Step-by-step explanation:

Given expressions:

        Pair 1 :  (x + 2)²  and x² + 4

        Pair 2: (x + 5)² and x² + 10x + 25

Problem;

Find if the expressions are equivalent

Solution:

i. For the first pair, the approach is to expand the expression in the parentheses;

           (x + 2)² = (x + 2)(x + 2)

                       = x² + 2x + 2x + 4

                       = x² + 4x + 4

Now, comparing x² + 4 and x² + 4x + 4

The two expressions are not equal;

      x² + 4x + 4  differs from x² + 4 by 4x

ii. Pair 2: (x + 5)² and x² + 10x + 25

      (x + 5)² = (x + 5) (x + 5)

                   = x² + 5x + 5x + 25

                   = x² + 10x + 25

The two expressions are;

    x² + 10x + 25  and  x² + 10x + 25