The side lengths of a rectangle are represented by the expressions $x + 2$ and $x - 5$. If the area of the rectangle is 60 square units, what is the perimeter?

Question

02-11-2026
Mathematics

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The side lengths of a rectangle are represented by the expressions $x + 2$ and $x - 5$. If the area of the rectangle is 60 square units, what is the perimeter?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T17:33:59-05:00

Set up the equation $ (x+2)(x-5) = 60 $ for the area of the rectangle.
Expand the equation to form the quadratic equation $ x^2 - 3x - 70 = 0 $.
Solve the quadratic equation using factoring to find $ x = 10 $, as it must satisfy $ x > 5 $.
Calculate the side lengths as 12 and 5, then determine the perimeter to be $ \boxed{34} $.

Explanation

Analyzing the problem and setting up the equation

We are given a rectangle with side lengths $x + 2$ and $x - 5$. The area of a rectangle is the product of its length and width, which is given as $60$ square units. Therefore, we have the equation: $(x + 2)(x - 5) = 60$. Additionally, side lengths must be positive, so we require $x + 2 > 0$ and $x - 5 > 0$, which together imply $x > 5$.

Expanding and forming the quadratic equation

First, we expand the product on the left side: $(x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10$. Setting this equal to the given area, we have $x^2 - 3x - 10 = 60$. Subtracting $60$ from both sides gives the quadratic equation $x^2 - 3x - 70 = 0$.

Solving the quadratic equation for x

To solve the equation $x^2 - 3x - 70 = 0$, we look for two numbers that multiply to $-70$ and add to $-3$. These numbers are $-10$ and $7$. Thus, the equation can be factored as $(x - 10)(x + 7) = 0$. This gives two potential solutions for $x$: $x = 10$ or $x = -7$. Since we established that $x > 5$, we must have $x = 10$.

Calculating the perimeter

With $x = 10$, the side lengths are $10 + 2 = 12$ and $10 - 5 = 5$. The perimeter $P$ of a rectangle is calculated using the formula $P = 2(\text{length} + \text{width})$. Substituting the side lengths, we get $P = 2(12 + 5) = 2(17) = 34$. Therefore, the perimeter of the rectangle is $34$ units.

Examples

When designing a room, you might know the required area and the difference between the lengths of the walls. For instance, if you want a rectangular room of 60 square feet where one side is 7 feet longer than the other (x+2 and x-5, where x=10), you can use this method to determine the exact dimensions and calculate the amount of baseboard needed for the perimeter. This helps in ordering materials precisely and ensures the floor plan fits within your available space.