https://file.aiquickdraw.com/mathbot/file/1726809652753ffgod1kz.jpg

The equation in the image is:

\[
x^2 = 8x - 31
\]

To solve this quadratic equation, let's rewrite it in standard form by moving all terms to one side:

\[
x^2 - 8x + 31 = 0
\]

Now, we can solve this using the quadratic formula. The quadratic formula is given by:

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

For the equation \(x^2 - 8x + 31 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -8\)
- \(c = 31\)

Now, let's calculate the discriminant:

\[
\Delta = b^2 - 4ac = (-8)^2 - 4(1)(31) = 64 - 124 = -60
\]

Since the discriminant is negative (\(\Delta = -60\)), the equation has no real solutions but has two complex solutions. 

We can now find the complex solutions:

\[
x = \frac{-(-8) \pm \sqrt{-60}}{2(1)} = \frac{8 \pm \sqrt{-60}}{2}
\]

Simplify the square root of \(-60\):

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

Now divide both terms by 2:

\[
x = 4 \pm \sqrt{15}i
\]

Thus, the solutions are:

\[
x = 4 + \sqrt{15}i \quad \text{or} \quad x = 4 - \sqrt{15}i
\]

Let me know if you'd like more details on any step! Here are some follow-up questions:

1. How does the discriminant affect the number and type of solutions in a quadratic equation?
2. What are the possible outcomes for the discriminant (positive, zero, or negative)?
3. What would the real solution be if the discriminant were zero?
4. Can you graph a quadratic equation with complex roots? How would it look?
5. How can the quadratic formula be derived from completing the square?

**Tip**: The quadratic formula works for any quadratic equation, but depending on the discriminant, it can give real or complex solutions.

Learn how to solve the quadratic equation x^2 = 8x - 31 using the quadratic formula. This problem involves algebraic manipulation and results in complex solutions due to a negative discriminant. Suitable for students in Grades 9-11.

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