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The image shows a problem related to the Pythagorean Theorem. It requires using the theorem to write a quadratic equation, then solve for \( x \), which represents the length of one of the legs of a walkway. Here's a step-by-step explanation:

### Problem:
Use the Pythagorean Theorem to write a quadratic equation in \( x \). Solve for \( x \), which represents one leg of the paved walkway.

### Given:
- The length of one leg is \( x \).
- The length of the other leg is \( (180 - x) \) meters.
- The length of the hypotenuse is \( 130 \) meters.

### Step 1: Applying the Pythagorean Theorem
The Pythagorean Theorem states:
\[
a^2 + b^2 = c^2
\]
Where:
- \( a = x \) (one leg),
- \( b = 180 - x \) (the other leg),
- \( c = 130 \) (the hypotenuse).

Substituting the values:
\[
x^2 + (180 - x)^2 = 130^2
\]

### Step 2: Expanding the equation
First, expand \( (180 - x)^2 \):
\[
(180 - x)^2 = 180^2 - 2(180)(x) + x^2 = 32400 - 360x + x^2
\]

Substitute this into the equation:
\[
x^2 + (32400 - 360x + x^2) = 16900
\]
Simplify:
\[
x^2 + x^2 + 32400 - 360x = 16900
\]
\[
2x^2 - 360x + 32400 = 16900
\]

### Step 3: Bring the equation into standard quadratic form
Move all terms to one side:
\[
2x^2 - 360x + 32400 - 16900 = 0
\]
\[
2x^2 - 360x + 15500 = 0
\]

### Step 4: Solve the quadratic equation
This is a quadratic equation in standard form:
\[
2x^2 - 360x + 15500 = 0
\]
We can solve this using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where:
- \( a = 2 \),
- \( b = -360 \),
- \( c = 15500 \).

Let me calculate the solution for you.The solutions to the quadratic equation \( 2x^2 - 360x + 15500 = 0 \) are:

\[
x_1 = 108.7 \, \text{m}, \quad x_2 = 71.3 \, \text{m}
\]

Since \( x \) represents the length of a leg, both values are valid, and they represent the possible lengths of the two legs of the walkway.

Would you like more details or clarifications on any step? Here are some related questions:

1. How does the Pythagorean Theorem work for non-right triangles?
2. Can we derive the quadratic formula from completing the square?
3. How do you interpret a negative discriminant in a quadratic equation?
4. What are the practical applications of solving quadratic equations in real-life geometry problems?
5. How does factoring compare to the quadratic formula in solving quadratic equations?

**Tip:** In geometry, always check if the given triangle is a right triangle before applying the Pythagorean theorem!

Use the Pythagorean Theorem to write a quadratic equation in x. Solve the equation to determine the length of the two legs of the paved walkway.

Learn how to apply the Pythagorean Theorem to write and solve a quadratic equation. This problem helps calculate the length of the legs of a paved walkway using algebra and geometry. Ideal for students in Grades 9-11.

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