Let $r$ and $s$ be the roots of $2x^2 + 5x - 13 = 0.$ Find $r^2 + s^2.$

To find $ r^2 + s^2 $, where $ r $ and $ s $ are the roots of the quadratic equation $ 2x^2 + 5x - 13 = 0 $, we can use the following relationships from Vieta's formulas:

\[
r + s = -\frac{b}{a} \quad \text{and} \quad rs = \frac{c}{a}
\]

For the given quadratic equation $ 2x^2 + 5x - 13 = 0 $, the coefficients are $ a = 2 $, $ b = 5 $, and $ c = -13 $.

1. **Sum of the roots**:
   \[
   r + s = -\frac{5}{2}
   \]

2. **Product of the roots**:
   \[
   rs = \frac{-13}{2}
   \]

We need to find $ r^2 + s^2 $. This can be expressed as:
\[
r^2 + s^2 = (r+s)^2 - 2rs
\]

Now, substitute the values of $ r+s $ and $ rs $:

\[
r^2 + s^2 = \left(-\frac{5}{2}\right)^2 - 2 \times \left(\frac{-13}{2}\right)
\]

Calculate each term:

\[
\left(-\frac{5}{2}\right)^2 = \frac{25}{4}
\]

\[
2 \times \left(\frac{-13}{2}\right) = -\frac{26}{2} = -13
\]

So,

\[
r^2 + s^2 = \frac{25}{4} + 13
\]

Convert 13 to a fraction with a denominator of 4:

\[
13 = \frac{52}{4}
\]

Thus:

\[
r^2 + s^2 = \frac{25}{4} + \frac{52}{4} = \frac{77}{4}
\]

Therefore, the value of $ r^2 + s^2 $ is:

\[
\boxed{\frac{77}{4}}
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the sum of the squares of the roots of $ x^2 + 4x + 5 = 0 $?
2. How do you find the sum and product of roots for a quadratic equation?
3. What is the value of $ r^2 + s^2 $ if $ r $ and $ s $ are roots of $ 3x^2 - 2x + 1 = 0 $?
4. How do you derive Vieta's formulas?
5. Can you express the sum of squares of the roots in terms of the coefficients of any quadratic equation?
6. How would the process change if the quadratic equation had complex roots?
7. What are the roots of the equation $ 2x^2 + 5x - 13 = 0 $?
8. What is the discriminant of the equation $ 2x^2 + 5x - 13 = 0 $ and what does it tell us?

**Tip:** When working with quadratic equations, Vieta's formulas provide a quick way to relate the roots to the coefficients without needing to solve for the roots explicitly.

Learn how to find r^2 + s^2 for the quadratic equation 2x^2 + 5x - 13 = 0 using Vieta's formulas. Follow a detailed step-by-step solution covering quadratic equations and Vieta's formulas. Suitable for students in Grades 9-12.

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