solve the quadratic equation x to the power of 2 -8x +24=4

The given quadratic equation is:

\[
x^2 - 8x + 24 = 4
\]

To solve this, first move all terms to one side by subtracting 4 from both sides:

\[
x^2 - 8x + 24 - 4 = 0
\]

Simplify:

\[
x^2 - 8x + 20 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

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

Here, \(a = 1\), \(b = -8\), and \(c = 20\).

First, calculate the discriminant:

\[
\Delta = b^2 - 4ac = (-8)^2 - 4(1)(20) = 64 - 80 = -16
\]

Since the discriminant is negative (\(\Delta = -16\)), the equation has two complex solutions. The solutions are:

\[
x = \frac{-(-8) \pm \sqrt{-16}}{2(1)} = \frac{8 \pm 4i}{2}
\]

Simplifying:

\[
x = 4 \pm 2i
\]

So, the solutions are:

\[
x = 4 + 2i \quad \text{and} \quad x = 4 - 2i
\]

These are the two complex solutions to the quadratic equation.

Would you like further details on complex numbers or the quadratic formula?

Here are 5 related questions for practice:
1. Solve \(x^2 + 6x + 10 = 0\) using the quadratic formula.
2. What happens when the discriminant is zero?
3. Can a quadratic equation have more than two roots? Why or why not?
4. Solve the equation \(x^2 - 6x + 9 = 0\).
5. What is the vertex form of a quadratic equation?

**Tip:** A negative discriminant indicates complex roots, which involve imaginary numbers (denoted by \(i\), where \(i^2 = -1\)).

This page provides a step-by-step solution to the quadratic equation x^2 - 8x + 24 = 4. Learn how to solve using the quadratic formula, handle a negative discriminant, and understand complex solutions. Suitable for students in Grades 9-12.

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