A rectangle has a length that is 5 inches greater than its width, and its area is 104 square inches. The equation (x + 5)x = 104 represents the situation, where x represents the width of the rectangle. The first step in solving by factoring is to write the equation in standard form, setting one side equal to zero. What is the equation for the situation, written in standard form?

Given the statement:

A rectangle has a length that is 5 inches greater than its width, and its area is 104 square inches. The equation (x + 5)x = 104 represents the situation, where x represents the width of the rectangle.

Determine the width of the rectangle.

Formula for the area of the rectangle:

A = lw

Solution:

In order to determine the width of the rectangle, we simply needed to distribute x to each value inside the parenthesis, and then transpose 104 to the other side of the equation. Hence, we must take note that in transposing a number, the sign changes.

(x + 5)x = 104

x^2 + 5x = 104

x^2 + 5x - 104 = 0

a = 1

b = 5

c = -104

Now to solve the value of x, we simply needed to apply the quadratic formula.

The Quadratic formula:

x = −b ± √(b^2 − 4ac)/2a

is used to solve quadratic equations where a ≠ 0, in the form
ax^2+bx+c=0

When b^2−4ac=0 there is one real root.

When b^2−4ac>0 there are two real roots.

When b^2−4ac<0 there are no real roots, only a complex number.

Substitute the given values of a, b and c to the quadratic formula.

x = −b ± √(b^2 − 4ac)/2a

x = −(5) ± √((5)^2 − 4(1)(-104))/2(1)

x = −5 ± √(25 + 416)/2

x = −5 ± √(441)/2

x = −5 ± 21/2

Solve for +- separately.

x = -5 + 21 / 2
x = 16/2
x = 8

x = - 5 - 21 / 2
x = -26
x = -13

Since we cannot have a negative width, therefore we only choose the positive answer, which in our case is 8.

Final answer:

Width = 8 inches