Question

Martha sells custom painted square canvases. She determined that the amount of tissue paper she'll need to wrap each canvas can be represented by

y= x^2 + 2x + 1, where y is the total square inches of tissue paper and x is the side length of the canvas in inches. Write and solve an equation to determine the side length of the canvas is Martha needs 49 square inches of tissue paper.

Answer 1

To find the side length of a canvas when 49 square inches of tissue paper is needed, the quadratic equation y = x^2 + 2x + 1 is solved with y=49, which simplifies to x^2 + 2x - 48 = 0 yielding a side length of 6 inches.

Step-by-step explanation:

To determine the side length of the canvas when 49 square inches of tissue paper is needed, we need to solve the quadratic equation provided: y = x^2 + 2x + 1. Given y=49, we replace y in the equation and solve for x:

49 = x^2 + 2x + 1

To solve it, we first subtract 49 from both sides:

0 = x^2 + 2x + 1 - 49

0 = x^2 + 2x - 48

This equation can be factored into:

(x + 8)(x - 6) = 0

Setting each factor equal to zero gives us two possible solutions:

• 
  
• x + 8 = 0 → x = -8
• 
  
• x - 6 = 0 → x = 6

Since a canvas side length cannot be negative, we disregard the negative solution. Therefore, the side length of the canvas is 6 inches.

Answer 2

Given:

The equation is

`y=x^2+2x+1`

where y is the total square inches of tissue paper and x is the side length of the canvas in inches.

To find:

The side length of the canvas is Martha needs 49 square inches of tissue paper.

Solution:

We have,

`y=x^2+2x+1`

Putting y=49, we get

`49=x^2+2x+1`

`0=x^2+2x+1-49`

`0=x^2+2x-48`

Splitting the middle term, we get

`0=x^2+8x-6x-48`

`0=x(x+8)-6(x+8)`

`0=(x+8)(x-6)`

Using zero product property, we get

`x+8=0` and
`x-6=0`

`x=-8` and
`x=6`

Side length cannot be negative. So, only possible value of x is 6.

Therefore, Martha needs 6 inches side length of the canvas for 49 square inches of tissue paper.