Question

What value of a will make the equation true?

(x+a)^2=x^2+16x+64
A. a=16
B. a=8
C. a=64
D. a=4

Answer 1

B. a=8

Explanation:

First, let's look at the expression that is on the right side of the equal sign:

`x^2+16x+64`

This expression is equal to:

`(x+a)^2`

So, it must mean that the expression on the right is merely the expanded form of the expression on the left.

Recall the expanded form for the square of a sum:

`(a+b)^2=a^2+b^2+2ab`

Notice that 64 is equal to 8 squared, and 16x is equal to 2 times 8 times x.

This must mean that
`a=8`.

Thus, our answer is:

B. a=8

Answer 2

To solve for the value of a that will make the equation true, we can expand the left side of the equation using the square of a binomial formula:

(x + a)^2 = x^2 + 2ax + a^2

Substituting this expression back into the original equation, we get:

x^2 + 2ax + a^2 = x^2 + 16x + 64

We can then simplify this equation by canceling out the x^2 terms on both sides, which gives:

2ax + a^2 = 16x + 64

Next, we can isolate the variable a on one side of the equation by subtracting 2ax and 64 from both sides:

a^2 - 2ax = 64 - 16x

Finally, we can factor out the variable a from the left side of the equation:

a(a - 2x) = 64 - 16x

To solve for a, we can divide both sides by (a - 2x):

a = (64 - 16x)/(a - 2x)

However, we must also note that the denominator (a - 2x) cannot be zero, as this would result in a division by zero error. Thus, we must ensure that a ≠ 2x.

Therefore, the value of a that will make the equation true is:

a = (64 - 16x)/(a - 2x), where a ≠ 2x.

Explanation: