Question

The rectangle below has an area of
\[x^2-7x+10\] square meters and a width of
\[x-5\] meters.
What expression represents the length of the rectangle?

Answer 1

To find the length of the rectangle, we can use the formula for the area of a rectangle, which is given by

\[
\text{Area} = \text{Length} \times \text{Width}.
\]

We are given that the area of the rectangle is

\[
x^2 - 7x + 10 \quad \text{(square meters)},
\]

and the width is

\[
x - 5 \quad \text{(meters)}.
\]

Let \( L \) represent the length of the rectangle. Then, we can express the area as

\[
L \times (x - 5) = x^2 - 7x + 10.
\]

To find the length \( L \), we can isolate \( L \) by dividing both sides of the equation by the width \( x - 5 \):

\[
L = \frac{x^2 - 7x + 10}{x - 5}.
\]

Next, we need to simplify the right-hand side. We can factor the quadratic expression \( x^2 - 7x + 10 \):

\[
x^2 - 7x + 10 = (x - 2)(x - 5).
\]

Now we substitute this factored form back into our expression for \( L \):

\[
L = \frac{(x - 2)(x - 5)}{x - 5}.
\]

As long as \( x - 5 \neq 0 \) (which means \( x \neq 5 \)), we can cancel \( x - 5 \) from the numerator and denominator:

\[
L = x - 2.
\]

Thus, the expression that represents the length of the rectangle is

\[
\boxed{x - 2} \quad \text{(meters)}.
\]