Question

Which term completes the product so that it is the difference of squares?

(−5x−3)(−5x+________)

−9
−3
3
9

Answer 1

To determine which term completes the product to form the difference of squares, we start by recognizing the structure of the difference of squares formula, which is:

\[
a^2 - b^2 = (a - b)(a + b)
\]

In the expression \((-5x - 3)(-5x + \_\_\_)\), we see that the first part (\(-5x\)) is the same, so we can denote it as \(a = -5x\). The second part requires us to find some value \(b\) such that the expression represents a difference of squares.

From the difference of squares formula, we can rewrite it as follows:

\[
(-5x)^2 - b^2 = 25x^2 - b^2
\]

To find \(b\), we note that \(b\) relates to the constant term, which corresponds to the term we need to fill in the blank. Since in the first factor (\(-5x - 3\)) the constant is \(-3\), we will use its absolute value to find \(b\). In order to maintain the difference of squares format, the other term of the second binomial must match the pattern of subtraction.

We can see that the respective symmetrical term that corresponds to the constant should be \(3\) since \(b^2\) will correspond to the square of this number, and subtracting \(3^2\) will give us the desired structure.

Thus, the term that completes the product is:

\[
3
\]

So, filling in the blank:

\[
(-5x - 3)(-5x + 3)
\]

To check, we can verify that:

\[
(-5x - 3)(-5x + 3) = (-5x)^2 - (3)^2 = 25x^2 - 9
\]

Hence, the correct term to complete the product so that it forms the difference of squares is:

\[
\boxed{3}
\]