Question

to find the area, a of a rectangle you multiply the rectangle’s length, l by its width, w. what is the width of a rectangle with the following dimensions:
a = x² + 7x + 10
l = (x + 2)

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = l \times w \), where \( l \) is the length and \( w \) is the width. We need to solve for \( w \), so we can rearrange the formula to \( w=\frac{A}{l} \).

Step2: Substitute the given expressions for \( A \) and \( l \)

We know that \( A = x^{2}+7x + 10 \) and \( l=(x + 2) \). So we need to perform the division \( w=\frac{x^{2}+7x + 10}{x + 2} \).

Step3: Factor the numerator

We factor the quadratic expression \( x^{2}+7x + 10 \). We need two numbers that multiply to \( 10 \) and add up to \( 7 \). The numbers are \( 2 \) and \( 5 \). So \( x^{2}+7x + 10=(x + 2)(x+5) \).

Step4: Simplify the fraction

Now we substitute the factored form of the numerator into the expression for \( w \): \( w=\frac{(x + 2)(x + 5)}{x + 2} \). Since \( x
eq - 2 \) (we can't divide by zero), we can cancel out the common factor \( (x + 2) \) in the numerator and the denominator. So \( w=x + 5 \).

Answer:

The width of the rectangle is \( x + 5 \)