Question

question
the width of a rectangle is the length minus 6 units. the area of the rectangle is 7 square units. what
is the length, in units, of the rectangle?

Explanation:

Step1: Set up the equation

Let the length of the rectangle be \(x\) units. Then the width is \(x - 6\) units. The area of a rectangle is \(A=\text{length}\times\text{width}\). So, \(x(x - 6)=7\).
Expanding gives \(x^{2}-6x - 7=0\).

Step2: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 1\), \(b=-6\), \(c = - 7\)), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-6)^{2}-4\times1\times(-7)=36 + 28=64\).
Then \(x=\frac{6\pm\sqrt{64}}{2}=\frac{6\pm8}{2}\).
We have two solutions: \(x_{1}=\frac{6 + 8}{2}=\frac{14}{2}=7\) and \(x_{2}=\frac{6-8}{2}=\frac{-2}{2}=-1\).
Since length cannot be negative, we discard \(x=-1\).

Answer:

\(7\)