Algebra 2 Quadratic equations (help)

Question

The length of a rectangle is 5 inches more than twice a number. The width is 3 inches less than the same number.If the area of the rectangle is 76 sq inches, what is the number?

Answer 1

Let "n" be "a number" then twice the number is "2n" and 5 more than that is "2n + 5" so:

Length = 2n + 5

3 less than the number is "n - 3" so:

Width = n - 3

Area = length x width

A = (2n + 5)(n - 3)

76 = (2n + 5)(n - 3)

76 = 2n² - 6n + 5n - 15

0 = 2n² - n - 15 - 76

2n² - n - 91 = 0

Sometimes it's not easy to realize a quadratic equation can be factored, so I'll use the Quadratic Formula:

For ax² + bx + c = 0

a = 2, b = -1, c = -91

x = -(-1) ± √((-1)2 - 4(2)(-91))/(2•2)

x = (1 ± √(1 + 728))/4

x = (1 ± √729)/4

Breaking this into 2 parts (the "plus" and the "minus"):

x = (1 + √729)/4 = (1 + 27)/4 = 28/4 = 7

x = (1 - √729)/4 = (1 - 27)/4 = -26/4 = -6.5

Plugging in n = 7 we get Length = 19 and Width = 4

Plugging in n = - 6.5 we get Length = -8 and Width = - 9.5 and obviously we can throw this out since length and width cannot be negative.

So the number is 7.

Answer 2

Hellos,

First you have to write an equation for the length and width of the rectangle. Let's say the number is x. Then:

Length = 2x+5

Width = x-3

And we know the area of a rectangle is (length * width) and equals 76.

Let us multiply the equations we have together:

(2x+5)(x-3)=76

-> 2x^2 -6x+5x-15=76. Put like terms together and you get

2x^2 -x -91=0. Now you can factor this out (13 *7=91) and equal them to zero to find the value of x.

(2x+13) = 0. When you solve this it gives you -6.5 which it is not possible to have a negative length.

(X-7) = 0. When you solve this it gives you X = 7. If you want to check your answer then multiply (2x+5)(x-3) with x =7 and make sure it equals 76.

Answer 3

Let x be the number, L be the length and W be the width.

L = 2x+5

W=x-3

Area, A = LW = 76 in²

A = (2x+5)(x-3) = 2x²-6x + 5x -15 = 2x²-x-15 =76

2x²-x-91 = 0

Need to solve for x

(2x+13)(x-7) = 0 --> x = 7, -13/2 Take the positive number

x=7

Answer 4

Let x = the number

Then length = 2x + 5 and width = x - 3

So, (2x+5)(x-3) = 76

2x² - x - 91 = 0

(2x + 13)(x - 7) = 0

x = -6.5 or x = 7

If x = -6.5, then the width and length would both be negative.

So, x = 7.