Solve x2 = 5x -7 using the quadratic formula. What can you conclude about the nature of the solution?

The given equation is:

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

To use the quadratic formula, we first need to rearrange the equation into standard quadratic form:

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

Now, the quadratic formula is given by:

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

For the equation \( x^2 - 5x + 7 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 7 \)

### Step 1: Calculate the discriminant
The discriminant \( \Delta \) is given by:

\[
\Delta = b^2 - 4ac
\]

Substituting the values:

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

### Step 2: Analyze the discriminant
The discriminant \( \Delta = -3 \) is negative, which means the quadratic equation has two complex (non-real) solutions.

### Step 3: Calculate the solutions
Now, substitute the values into the quadratic formula:

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

Since \( \sqrt{-3} \) can be written as \( i\sqrt{3} \) (where \( i \) is the imaginary unit), we have:

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

### Conclusion:
The quadratic equation has two complex (conjugate) solutions:
\[
x = \frac{5 + i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{5 - i\sqrt{3}}{2}
\]

These solutions are not real because the discriminant is negative.

Would you like further details or have any questions about this solution?

Here are 5 related questions:
1. How do complex solutions differ from real solutions?
2. What happens when the discriminant is zero?
3. Can a quadratic equation have more than two solutions?
4. How can we visually interpret complex solutions in a graph?
5. What real-life situations might involve complex numbers?

**Tip:** When the discriminant is negative, the solutions will always involve complex numbers with imaginary parts.

Learn how to solve the quadratic equation x^2 = 5x - 7 step-by-step using the quadratic formula. Explore the nature of solutions involving complex numbers and understand the concept of discriminant. Suitable for high school students.

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