Consecutive Integer Problems - Quadratic

Question

The length and width of a rectangle are consecutive integers. The area of the rectangle is 90 square feet. Find the length and width of the rectangle.

Answer 1

A = x(x + 1)

90 = x² + x

0 = x² + x - 90

Can you solve for x and answer?

Answer 2

Hi Mackenzie P

x, and x + 1 represent two consecutive integers

9, 10

Please see below

Let the Length = x

Let the Width = x + 1

Since then area of a Rectangle = (Length)(Width) = 90

We have

x(x+1)=90

x² + x = 90

We can obtain a Quadratic Equation by subtracting 90 from both sides

x² + x - 90 = 0

This can be factored

(x +10)(x - 9) = 0

Length cannot be negative so only (x - 9) is the only useful factor

(x - 9) = 0

x = 9

The Length is 9

The Width x + 1= 9 + 1 =10

I hope this helps.

Answer 3

Hi!

Let's set l=length and w=width. We know that l and w are consecutive integers, which means that w=l+1.

The area of the rectangle is length*width = l*w = 90 sqft

We replace w into l*w = 90 => l(l+1)=90 => l²+l = 90.

To find l, we solve the equation for l.

l²+l = 90 => l²+l -90 = 0 => (l-9)(l+10)=0 => l=9,-10.

Since l is the length of a rectangle it must be positive. It follows that l=9 and w=9+1=10.

I hope this helps. Please get in touch if you have any questions. Good luck!