algebra question

Question

hte length of a rectangle is 12 m longer than its width and the area of the rectangle is 460 m2. Find the exact dimensions of the rectangle.

Answer 1

Hello , the answer for the question is the following:

Below is an explanation on how i get to the answer:

The given informations is the length is 12m longer than the width of the rectangle ( L=w+12)

As we know the area of a rectangle is Area= L*W (we will substitute the L with W+12 )

W*L = 460 m².

W*(W+12)=460m²

W²+12W-460 = 0, which is a quadratic equation. To find W, we apply the Completing the square method or quadratic formula:

here is how we complete he square:

(w+6)² -36-460=0

(w+6)²=496. apply the square root rule on both sides of the equation

W+6=±√496=±4√31, subtract 6 on both sides, so W = -6 + 4√31

length L = W + 12 = -6 + 4√31 + 12 = 6 + 4√31.

Answer 2

Good afternoon and thanks for the question!

The answer is W = -6 + 4√31 m and L = 6 + 4√31.

Below is an explanation of how I arrived at this answer.

We know that the length is 12m longer than the width of the rectangle. Therefore, we can represent the length in terms of the width with the following expression: L=W+12.

From Geometry, we know the area of a rectangle is L*W, so the area of this rectangle is:

(W+12)*W = 460 m².

Simplifying the left hand side of the above equation and subtracting 460 from both sides of the equation, we get:

W²+12W-460 = 0, which is a quadratic equation. To find W, we apply the quadratic formula:

W = (-12 ± √12²-4*1*(-460))/2 = (-12 ±√1984)/2 = (-12 ± √64*31)/2 = (-12 ± 8√31)/2 = -6 ± 4√31.

Because we are dealing with length, we can ignore the negative root of the quadratic equation, so W = -6 + 4√31 and length is L = W + 12 = -6 + 4√31 + 12 = 6 + 4√31. .