Question

What is the missing term? 100x2−36=4(5x+?)(5x−3)

Answer 1

Let's start by simplifying the left side of the equation:

\[
100x^2 - 36
\]

This expression can be factored using the difference of squares:

\[
100x^2 - 36 = (10x)^2 - 6^2 = (10x - 6)(10x + 6)
\]

Now we rewrite the equation:

\[
(10x - 6)(10x + 6) = 4(5x + ?)(5x - 3)
\]

Next, we simplify the right side:

\[
4(5x + ?)(5x - 3)
\]

We need to find the missing term denoted by "?". To do that, let's analyze the comparison of both sides as we expand the right side.

The expression \( (5x + ?)(5x - 3) \) can be expanded:

\[
(5x + ?)(5x - 3) = 25x^2 - 15x + ?(5x) - 3?
\]
\[
= 25x^2 + (? \cdot 5 - 15)x - 3?
\]

Now multiply this entire expression by \( 4 \):

\[
4(25x^2 + (? \cdot 5 - 15)x - 3?) = 100x^2 + 4(? \cdot 5 - 15)x - 12?
\]

Now we need to match this with the left side, which we factored as \( (10x - 6)(10x + 6) \). The coefficient of \( x^2 \) matches (both are \( 100x^2 \)), so we will set the corresponding linear terms on both sides equal.

On the left side, the linear terms are 0 since it is fully factored into two quadratic expressions. Therefore, we have:

\[
4(? \cdot 5 - 15) = 0
\]

This leads to:

\[
? \cdot 5 - 15 = 0
\]
\[
? \cdot 5 = 15
\]
\[
? = \frac{15}{5} = 3
\]

Now we can confirm our conclusions:

Substituting \( ? \) back into our original equation gives us:

\[
4(5x + 3)(5x - 3)
\]

We can also check the constant terms:

The constant term from the left side after factoring was \( -36 \).

Calculating the constant from the right side:

\[
4 \cdot 3 \cdot (-3) = -36
\]

Both sides match, thus confirming our value of \( ? \).

The missing term is:

\[
\boxed{3}
\]