Solve the equation using the quadratic formula.

xsquared2equals=6xminus−34




To solve the quadratic equation using the quadratic formula, let's first rewrite the equation in standard form:

\[
x^2 - 6x + 34 = 0
\]

The quadratic formula is given by:

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

For the equation \(x^2 - 6x + 34 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -6\)
- \(c = 34\)

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

\[
\Delta = b^2 - 4ac
\]
\[
\Delta = (-6)^2 - 4(1)(34) = 36 - 136 = -100
\]

### Step 2: Apply the quadratic formula
Since the discriminant is negative (\(\Delta = -100\)), the equation has complex (imaginary) solutions. Let's compute the solutions:

\[
x = \frac{-(-6) \pm \sqrt{-100}}{2(1)} = \frac{6 \pm \sqrt{-100}}{2}
\]
\[
x = \frac{6 \pm 10i}{2}
\]
\[
x = 3 \pm 5i
\]

### Final Answer:
The solutions to the equation are:
\[
x = 3 + 5i \quad \text{or} \quad x = 3 - 5i
\]

Would you like a detailed explanation of any step or more examples?

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Here are 5 related questions for deeper understanding:
1. What happens when the discriminant of a quadratic equation is positive?
2. How can you solve quadratic equations with real solutions using the quadratic formula?
3. What are complex numbers, and how are they used in equations with negative discriminants?
4. Can a quadratic equation have both real and complex solutions?
5. How do you graph the solutions of a quadratic equation on the complex plane?

**Tip:** The discriminant of a quadratic equation helps you quickly determine the nature of the roots—whether they are real, repeated, or complex.

Solve the equation using the quadratic formula.

x^2 - 6x + 34 = 0

Discover how to solve the quadratic equation x^2 - 6x + 34 using the quadratic formula, with detailed steps. Explore how to handle equations with negative discriminants, resulting in complex solutions. Ideal for students in Grades 9-11 studying algebra and complex numbers.

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