apply Pythagorean theorem then solve for x. (x-1)^2+(x)^2=(5)^2

Let's solve the equation using the Pythagorean theorem:

\[
(x - 1)^2 + x^2 = 5^2
\]

### Step 1: Expand the squares.

1. Expand \((x - 1)^2\):
\[
(x - 1)^2 = x^2 - 2x + 1
\]

2. Expand \(x^2\):
\[
x^2 = x^2
\]

3. The right-hand side is \(5^2\), which simplifies to 25:
\[
5^2 = 25
\]

### Step 2: Substitute and combine like terms.

Substitute the expanded expressions into the equation:
\[
(x^2 - 2x + 1) + x^2 = 25
\]

Combine the like terms:
\[
2x^2 - 2x + 1 = 25
\]

### Step 3: Move all terms to one side of the equation.

Subtract 25 from both sides:
\[
2x^2 - 2x + 1 - 25 = 0
\]

Simplify:
\[
2x^2 - 2x - 24 = 0
\]

### Step 4: Simplify the equation.

Divide the entire equation by 2 to simplify:
\[
x^2 - x - 12 = 0
\]

### Step 5: Solve the quadratic equation.

Use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = -1\), and \(c = -12\):

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-12)}}{2(1)}
\]

Simplify further:

\[
x = \frac{1 \pm \sqrt{1 + 48}}{2}
\]
\[
x = \frac{1 \pm \sqrt{49}}{2}
\]
\[
x = \frac{1 \pm 7}{2}
\]

This gives two possible solutions:

\[
x = \frac{1 + 7}{2} = \frac{8}{2} = 4
\]

\[
x = \frac{1 - 7}{2} = \frac{-6}{2} = -3
\]

### Final Answer:

The solutions for \(x\) are:

\[
x = 4 \quad \text{or} \quad x = -3
\]

Would you like further details or have any questions?

Here are 8 related questions:

1. How do you derive the quadratic formula?
2. Can you verify the solutions \(x = 4\) and \(x = -3\) by substituting them back into the original equation?
3. How is the Pythagorean theorem used in geometry?
4. What does the discriminant tell you about the nature of the roots in a quadratic equation?
5. How do you factorize the quadratic equation \(x^2 - x - 12 = 0\)?
6. What are the real-life applications of quadratic equations?
7. How do you graph the equation \(x^2 - x - 12 = 0\)?
8. How does changing the coefficients in a quadratic equation affect its graph?

**Tip:** Always check both potential solutions in the original equation to confirm they are correct.

Learn how to apply the Pythagorean theorem to solve the equation (x-1)^2 + x^2 = 25. This problem covers key concepts in algebra and geometry, including the Pythagorean theorem and quadratic equations. Suitable for students in Grades 9-12.

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