Question

a rectangle has an area of 35 square feet. its width is x+3 feet and the length is x+5 feet. write and solve an equation (and type the equation into the box below) to determine the length and width of the garden. please show all work: your original function, how you simplified so that you could refactor, what x is, and how you found the length and width.

Explanation:

Step1: Recall area formula for rectangle

The area \( A \) of a rectangle is given by \( A=\text{length} \times \text{width} \). We know the width is \( x + 3 \), the length is \( x + 5 \), and the area is 35 square feet. So we set up the equation: \( (x + 3)(x + 5)=35 \).

Step2: Expand the left - hand side

Using the FOIL method, \( (x + 3)(x + 5)=x^{2}+5x+3x + 15=x^{2}+8x + 15 \). So our equation becomes \( x^{2}+8x + 15 = 35 \).

Step3: Simplify to standard quadratic form

Subtract 35 from both sides to get \( x^{2}+8x+15 - 35=0 \), which simplifies to \( x^{2}+8x - 20 = 0 \).

Step4: Factor the quadratic equation

We need two numbers that multiply to - 20 and add to 8. The numbers are 10 and - 2. So we can factor the quadratic as \( (x + 10)(x - 2)=0 \).

Step5: Solve for x

Set each factor equal to zero: \( x+10 = 0 \) or \( x - 2=0 \). Solving \( x+10 = 0 \) gives \( x=-10 \), and solving \( x - 2=0 \) gives \( x = 2 \). Since length and width cannot be negative, we discard \( x=-10 \). So \( x = 2 \).

Step6: Find length and width

Width: Substitute \( x = 2 \) into \( x + 3 \), we get \( 2+3=5 \) feet.
Length: Substitute \( x = 2 \) into \( x + 5 \), we get \( 2+5=7 \) feet.

Answer:

The equation is \((x + 3)(x + 5)=35\) (or \(x^{2}+8x - 20 = 0\) after simplification), \(x = 2\), width is 5 feet, length is 7 feet.