Question

14 marie’s office is rectangular in shape and has an area of 250 square feet. the width of the office is (2x + 5) feet and the length is (3x − 1) feet. write a quadratic equation that represents this situation.
\\( x^2 - 13x - 250 = 0 \\)
\\( 6x^2 - 13x - 255 = 0 \\)
\\( 6x^2 + 13x - 255 = 0 \\)
\\( 5x^2 + 17x - 255 = 0 \\)

Explanation:

Step1: Recall the area formula of a rectangle

The area \(A\) of a rectangle is given by the formula \(A=\text{length}\times\text{width}\). Here, the width is \((2x + 5)\) feet and the length is \((3x-1)\) feet, and the area \(A = 250\) square feet. So we can write the equation as \((2x + 5)(3x - 1)=250\).

Step2: Expand the left - hand side

Using the FOIL method \((a + b)(c + d)=ac+ad+bc+bd\), where \(a = 2x\), \(b = 5\), \(c = 3x\), \(d=-1\):

$$
LATEXBLOCK0
$$

Step3: Set up the quadratic equation

Since \((2x + 5)(3x - 1)=250\) and \((2x + 5)(3x - 1)=6x^{2}+13x - 5\), we have:
\(6x^{2}+13x-5 = 250\)
Subtract 250 from both sides of the equation:
\(6x^{2}+13x-5 - 250=0\)
Simplify to get:
\(6x^{2}+13x-255 = 0\)

Answer:

\(6x^{2}+13x - 255 = 0\)