Solve the equation for all real solutions in simplest form.
5, k, squared, minus, 17, k, plus, 13, equals, 0
5k 
2
 −17k+13=
0

The equation given is:

\[
5k^2 - 17k + 13 = 0
\]

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 5 \), \( b = -17 \), and \( c = 13 \).

To solve it, we will use the **quadratic formula**:

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

### Step 1: Identify the coefficients
- \( a = 5 \)
- \( b = -17 \)
- \( c = 13 \)

### Step 2: Apply the quadratic formula

\[
k = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(5)(13)}}{2(5)}
\]

Simplify the discriminant:

\[
k = \frac{17 \pm \sqrt{289 - 260}}{10}
\]

\[
k = \frac{17 \pm \sqrt{29}}{10}
\]

### Step 3: Write the two solutions

\[
k_1 = \frac{17 + \sqrt{29}}{10}, \quad k_2 = \frac{17 - \sqrt{29}}{10}
\]

These are the two real solutions in simplest form.

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. What does the discriminant tell us about the nature of the solutions?
2. How do we graph a quadratic equation?
3. Can we factor this quadratic equation instead of using the quadratic formula?
4. What are the real-world applications of quadratic equations?
5. How does the quadratic formula work when the discriminant is negative?

**Tip:** The discriminant \(b^2 - 4ac\) helps you quickly determine if a quadratic equation has real or complex solutions.

Solve the equation for all real solutions in simplest form. 5k^2 − 17k + 13 = 0

Learn how to solve the quadratic equation 5k^2 − 17k + 13 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and understanding discriminants. Suitable for students in Grades 9-10.

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