The given equation is:

\[
x^2 - 8x = -1
\]

To solve this quadratic equation, follow these steps:

### Step 1: Move all terms to one side of the equation.
Add \(1\) to both sides to make the equation equal to \(0\):

\[
x^2 - 8x + 1 = 0
\]

### Step 2: Solve the quadratic equation using the quadratic formula.
The quadratic formula is:

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

For the equation \(x^2 - 8x + 1 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -8\)
- \(c = 1\)

Now, substitute these values into the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(1)}}{2(1)}
\]

Simplify:

\[
x = \frac{8 \pm \sqrt{64 - 4}}{2}
\]
\[
x = \frac{8 \pm \sqrt{60}}{2}
\]
\[
x = \frac{8 \pm \sqrt{4 \times 15}}{2}
\]
\[
x = \frac{8 \pm 2\sqrt{15}}{2}
\]

### Step 3: Simplify the result.
\[
x = 4 \pm \sqrt{15}
\]

Thus, the solutions are:

\[
x_1 = 4 + \sqrt{15}
\quad \text{and} \quad
x_2 = 4 - \sqrt{15}
\]

These are the two roots of the equation.

Do you need further details or have any questions?

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Here are 5 related questions:
1. How do you derive the quadratic formula?
2. Can you explain how to complete the square for a quadratic equation?
3. What are the properties of the discriminant in the quadratic formula?
4. How can you factor quadratic equations if possible?
5. What are the different methods for solving quadratic equations?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) to determine whether the roots are real or complex before solving.

Learn how to solve the quadratic equation x^2 - 8x = -1 with a step-by-step guide using the quadratic formula. This problem covers important algebraic concepts, including the use of the quadratic formula and equation transformation. Suitable for students in Grades 8-10.

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