Question

A rectangular field has dimensions of (2x-3) yards and (x+4) yards. the area of the field is 285 square yards. what are the length and width of the field?

Answer 1

The area of a rectangle is given by the multiplication of its dimensions:

$\text{Area} = \text{width} \times \text{length}$

In this case, we have an exact value for the area, and two variable expressions for width and length. Let's replace the formula above with what we're given:

$285 = (2x-3)(x+4)$

If you expand the right hand side, you have

$285 = 2 x^2+ 5 x-12$

And if you move all terms to the right hand side, this becomes

$0 = 2 x^2+ 5 x-297$

This is a quadratic equation, since it is in the form $ax^2+bx+c=0$, where $a = 2,\ b=5 \text{ and } c = -297$

If you plug these values in the generic formula

$x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

you get

$x = \dfrac{-27}{2} \text{ or } x = 11$

Let's see which dimensions they yield:

$x = \dfrac{-27}{2} \implies 2x-3 = -30,\quad x+4 = \dfrac{-19}{2}$

But the dimensions of a rectangle can't be negative, so we can't accept this answer.

The other solution yields

$x = 11 \implies 2x-3 = 19,\quad x+4 = 15$

So these are the dimensions of the rectangle