The quadratic equation that models the situation is x^2 +27x−160=0.
Let's denote the increase in both the length and width of the patio by the same amount as x. The original dimensions of the patio are 15 ft by 12 ft, so the new dimensions will be
15+x ft by 12+x ft.
The area of the original patio is given by the product of its length and width:
Original Area=15×12
The area of the expanded patio is the sum of the original area and the increase in area (160 ft²):
Expanded Area=Original Area+160
Now, let's set up the equation:
15×12+160=(15+x)×(12+x)
Simplify this equation to get the quadratic equation:
180+160=(15+x)(12+x)
340=180+27x+x^2
Subtract 180 from both sides:
160=27x+x^2
Now, rearrange the terms to get the quadratic equation in standard form:
x^2+27x−160=0
So, the quadratic equation that models the situation is x^2 +27x−160=0.