Math Problem Statement

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.

Solution

The image shows a problem related to the Pythagorean Theorem. It requires using the theorem to write a quadratic equation, then solve for xx, 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 xx. Solve for xx, which represents one leg of the paved walkway.

Given:

  • The length of one leg is xx.
  • The length of the other leg is (180x)(180 - x) meters.
  • The length of the hypotenuse is 130130 meters.

Step 1: Applying the Pythagorean Theorem

The Pythagorean Theorem states: a2+b2=c2a^2 + b^2 = c^2 Where:

  • a=xa = x (one leg),
  • b=180xb = 180 - x (the other leg),
  • c=130c = 130 (the hypotenuse).

Substituting the values: x2+(180x)2=1302x^2 + (180 - x)^2 = 130^2

Step 2: Expanding the equation

First, expand (180x)2(180 - x)^2: (180x)2=18022(180)(x)+x2=32400360x+x2(180 - x)^2 = 180^2 - 2(180)(x) + x^2 = 32400 - 360x + x^2

Substitute this into the equation: x2+(32400360x+x2)=16900x^2 + (32400 - 360x + x^2) = 16900 Simplify: x2+x2+32400360x=16900x^2 + x^2 + 32400 - 360x = 16900 2x2360x+32400=169002x^2 - 360x + 32400 = 16900

Step 3: Bring the equation into standard quadratic form

Move all terms to one side: 2x2360x+3240016900=02x^2 - 360x + 32400 - 16900 = 0 2x2360x+15500=02x^2 - 360x + 15500 = 0

Step 4: Solve the quadratic equation

This is a quadratic equation in standard form: 2x2360x+15500=02x^2 - 360x + 15500 = 0 We can solve this using the quadratic formula: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Where:

  • a=2a = 2,
  • b=360b = -360,
  • c=15500c = 15500.

Let me calculate the solution for you.The solutions to the quadratic equation 2x2360x+15500=02x^2 - 360x + 15500 = 0 are:

x1=108.7m,x2=71.3mx_1 = 108.7 \, \text{m}, \quad x_2 = 71.3 \, \text{m}

Since xx 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!

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Quadratic Equations

Formulas

Pythagorean Theorem: a^2 + b^2 = c^2
Quadratic Equation: ax^2 + bx + c = 0
Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Pythagorean Theorem
Quadratic Formula

Suitable Grade Level

Grades 9-11