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The given equation is \( x^2 = 5x - 7 \).

To use the quadratic formula, we need to rewrite this equation in the standard quadratic form, which is:

\[
ax^2 + bx + c = 0
\]

So, we can rearrange the equation as:

\[
x^2 - 5x + 7 = 0
\]

Now we have:
- \(a = 1\)
- \(b = -5\)
- \(c = 7\)

The quadratic formula is given by:

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

### Step 1: Calculate the discriminant (\( \Delta \)):

\[
\Delta = b^2 - 4ac = (-5)^2 - 4(1)(7) = 25 - 28 = -3
\]

### Step 2: Use the quadratic formula:

Since the discriminant is negative (\( \Delta = -3 \)), the solutions will be complex. Let’s proceed with the formula:

\[
x = \frac{-(-5) \pm \sqrt{-3}}{2(1)} = \frac{5 \pm \sqrt{-3}}{2}
\]

We simplify further:

\[
x = \frac{5 \pm i\sqrt{3}}{2}
\]

Thus, the solutions are:

\[
x_1 = \frac{5 + i\sqrt{3}}{2}, \quad x_2 = \frac{5 - i\sqrt{3}}{2}
\]

### Nature of the solution:
Since the discriminant is negative, the solutions are **complex** (non-real) numbers. These solutions come in conjugate pairs due to the presence of \(i\), the imaginary unit.

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Would you like a more detailed explanation of any part of the solution?

Here are 5 questions to extend your understanding:

1. How does the discriminant determine the nature of the solutions in a quadratic equation?
2. What happens if the discriminant is positive or zero?
3. How do we handle complex numbers in practical applications?
4. Can you graph the solutions of this equation, even though they are complex?
5. What is the geometric interpretation of complex solutions?

**Tip:** When solving quadratic equations, always check the discriminant first to understand the nature of the solutions (real, complex, or repeated).

Solve x^2 = 5x - 7 using the quadratic formula. What can you conclude about the nature of the solution?

Solve the quadratic equation x^2 = 5x - 7 with a step-by-step solution using the quadratic formula. Learn about discriminants and understand why the solutions are complex numbers. Suitable for students in Grades 9-12.

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