Question

A photograph has a length that is 3
inches longer than its width, x. So its area is given by the expression x(x+3)
square inches. If the area of the photograph is 70
square inches, what is the length of the photograph?

The length of the photograph is blank inches.

Answer 1

To find the length of the photograph, we need to first set up the equation based on the information provided.

1. The width of the photograph is \( x \) inches.
2. The length of the photograph, being 3 inches longer than the width, is \( x + 3 \) inches.
3. The area of the photograph is given by the expression \( x(x + 3) \) square inches.

According to the problem, the area is also equal to 70 square inches. Therefore, we can set up the equation:

\[
x(x + 3) = 70
\]

Expanding this equation, we get:

\[
x^2 + 3x = 70
\]

Now, we move 70 to the left side to set the equation to zero:

\[
x^2 + 3x - 70 = 0
\]

Next, we can factor this quadratic equation. We are looking for two numbers that multiply to \(-70\) and add to \(3\). The numbers \(10\) and \(-7\) fit this description:

\[
(x + 10)(x - 7) = 0
\]

Setting each factor to zero gives us:

\[
x + 10 = 0 \quad \text{or} \quad x - 7 = 0
\]
\[
x = -10 \quad \text{or} \quad x = 7
\]

Since \( x \) represents a length, it must be positive, so we take:

\[
x = 7
\]

Now we can find the length of the photograph:

\[
\text{Length} = x + 3 = 7 + 3 = 10 \text{ inches}
\]

Thus, the length of the photograph is:

\[
\boxed{10} \text{ inches}
\]