To find the quadratic equation that models the situation, let's break down the process step by step.
### Step-by-Step Solution:
1. Current Dimensions:
- The current length of the patio [tex]\( l \)[/tex] is 15 feet.
- The current width of the patio [tex]\( w \)[/tex] is 12 feet.
2. Initial Area Calculation:
- The initial area [tex]\( A \)[/tex] of the patio is calculated as:
[tex]\[
A = l \times w = 15 \times 12 = 180 \text{ square feet}
\][/tex]
3. Area Increase:
- The patio's area is to be increased by 160 square feet.
- Therefore, the new total area after expansion will be:
[tex]\[
A_{\text{new}} = 180 + 160 = 340 \text{ square feet}
\][/tex]
4. Introducing the Extension:
- Let [tex]\( x \)[/tex] be the length by which both the length and the width are extended.
- The new dimensions will be:
[tex]\[
\text{New Length} = 15 + x
\][/tex]
[tex]\[
\text{New Width} = 12 + x
\][/tex]
5. Form the Equation:
- The new area should be equal to the total area. So we set up the following equation:
[tex]\[
(15 + x)(12 + x) = 340
\][/tex]
6. Expanding the Equation:
- Expand the left side:
[tex]\[
15 \cdot 12 + 15 \cdot x + 12 \cdot x + x^2 = 340
\][/tex]
- Simplify the expression:
[tex]\[
180 + 15x + 12x + x^2 = 340
\][/tex]
- Combine the like terms:
[tex]\[
x^2 + 27x + 180 = 340
\][/tex]
7. Form the Quadratic Equation:
The equation is now in a standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
x^2 + 27x + 180 - 340 = 0
\][/tex]
8. Final Simplification:
- Simplify the constants:
[tex]\[
x^2 + 27x - 160 = 0
\][/tex]
### Conclusion:
The quadratic equation that best models the situation is:
[tex]\[
(x + 15)(x + 12) = 340
\][/tex]
Which simplifies to:
[tex]\[
x^2 + 27x - 160 = 0
\][/tex]
From the options provided, the correct quadratic equation is:
[tex]\[
\boxed{(15 + x)(12 + x) = (15 \times 12) + 160}
\][/tex]
This corresponds to the step-by-step explanation and the equation [tex]\( x^2 + 27x - 160 = 0 \)[/tex].
