x^2-6x+10 solve for x using the quadratic equation

To solve the quadratic equation \( x^2 - 6x + 10 = 0 \) using the quadratic formula, we follow these steps:

The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the given equation \( x^2 - 6x + 10 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -6 \)
- \( c = 10 \)

### Step 1: Calculate the discriminant
The discriminant \( \Delta \) is:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
\Delta = (-6)^2 - 4(1)(10) = 36 - 40 = -4
\]
The discriminant is \( -4 \), which is negative. This means the solutions will be complex numbers.

### Step 2: Calculate the roots
Now, using the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{-4}}{2(1)}
\]
Simplifying further:
\[
x = \frac{6 \pm \sqrt{-4}}{2}
\]
Since \( \sqrt{-4} = 2i \) (where \( i \) is the imaginary unit), the equation becomes:
\[
x = \frac{6 \pm 2i}{2}
\]
Finally, dividing both terms by 2:
\[
x = 3 \pm i
\]

### Final Answer:
The solutions to the equation are:
\[
x = 3 + i \quad \text{or} \quad x = 3 - i
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you solve quadratic equations with real roots?
2. What does the discriminant tell us about the nature of roots?
3. How do you graph a quadratic function with complex roots?
4. Can you explain how the imaginary unit \( i \) is defined?
5. How would you factor a quadratic equation if it has real roots?

**Tip:** When solving quadratic equations, always check the discriminant first to determine whether the roots are real or complex.

Learn how to solve the quadratic equation x^2 - 6x + 10 step-by-step with the quadratic formula. Understand the concept of discriminant and complex roots. Suitable for high school students (Grades 10-12).

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