To find the zeros of the quadratic function \(x^2 - 4x + 6\), we need to solve the equation \(x^2 - 4x + 6 = 0\).

We can use the quadratic formula:

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

For the given equation \(x^2 - 4x + 6 = 0\):

- \(a = 1\)
- \(b = -4\)
- \(c = 6\)

Plugging these values into the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(6)}}{2(1)}
\]

Simplifying the expression:

\[
x = \frac{4 \pm \sqrt{16 - 24}}{2}
\]

\[
x = \frac{4 \pm \sqrt{-8}}{2}
\]

Since \(\sqrt{-8}\) is an imaginary number, we can express it as \(2i\sqrt{2}\):

\[
x = \frac{4 \pm 2i\sqrt{2}}{2}
\]

\[
x = 2 \pm i\sqrt{2}
\]

So, the zeros of the quadratic function are:

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

These are complex numbers.

Would you like more details or have any questions about this solution? Here are some related questions you might find interesting:

1. How do you determine if a quadratic equation has real or complex roots?
2. Can complex roots occur in pairs? Why?
3. What is the significance of the discriminant in the quadratic formula?
4. How would you graph the function \(x^2 - 4x + 6\) and interpret the complex roots?
5. How does the quadratic formula handle cases where the discriminant is zero?

**Tip:** The discriminant \(\Delta = b^2 - 4ac\) in the quadratic formula determines the nature of the roots: if \(\Delta > 0\), the roots are real and distinct; if \(\Delta = 0\), the roots are real and equal; if \(\Delta < 0\), the roots are complex.

Discover how to find the zeros of the quadratic equation x^2 - 4x + 6 using the quadratic formula. This detailed solution explores the discriminant and explains why the roots are complex numbers. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation