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: Factor the quadratic equation

For the quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 1\), \(b=-6\), \(c = - 7\)), we use the formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or factor.
We factor \(x^{2}-6x - 7=(x - 7)(x+1)=0\).

Step3: Solve for \(x\)

Set each factor equal to zero:
\(x - 7=0\) gives \(x = 7\); \(x+1=0\) gives \(x=-1\).
Since length cannot be negative, we discard \(x=-1\).

Answer:

\(7\)