Perfect Square Expressions

Question

In each equation, what expression could be substituted for a so the equation is true for all values of x?

1. x^2 = a^2
2. (3x)^2 = a^2 - 9x^2
3. a^2 = 7x • 7x - 7x^2

Answer 1

Hi Cameron,

I hope that you are doing well during this time, and that this message finds you in a good place.

To start these problems, you want to think of what you could replace "a" with to satisfy the expression.

1. x² = a²

Here the exponents are the same on both sides of the equal sign; so you could just as easily consider this problem without any exponents, as just saying:

x = a

In this case the only that you could replace a with, and satisfy the expression, is "x"

So, for 1, the answer would be "a" can be replaced by "x"

The second can be rewritten as:

9x² = a² - 9x²

Since (3x)² = 9x²

Here you have to consider the following expression:

9 = a - 9

Solving for a by adding 9 to both sides:

a = 18

So, for the second problem, "a" must be replaced by "√18⋅x" or "3√2⋅x" as the (√18 ⋅x)² = 18x², which satisfies the expression.

The third problem can be simplified to:

a² = 49x² - 7x²

Further simplifying:

a² = 42x²

The answer to the third question would be replacing "a" with "√42⋅x" as (√42⋅x)² = 42x², which satisfies the expression.

I hope that this helps. Feel free to reach out, if you need any more help!

Best,

Ray