https://file.aiquickdraw.com/mathbot/file/172592721381603lf7ejg.jpg

The equation in the image is:

\[
9r^2 - 5r - 10 = 0
\]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where:

- \( a = 9 \)
- \( b = -5 \)
- \( c = -10 \)

We can solve this quadratic equation using the quadratic formula:

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

Substituting the values for \(a\), \(b\), and \(c\):

\[
r = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(9)(-10)}}{2(9)}
\]

Let's solve this step-by-step.The solutions to the quadratic equation \(9r^2 - 5r - 10 = 0\) are approximately:

\[
r_1 \approx 1.368 \quad \text{and} \quad r_2 \approx -0.812
\]

Let me know if you'd like further clarification or details on the solution!

Here are some related questions:
1. How would you verify these solutions using factoring?
2. What is the discriminant of a quadratic equation, and how does it affect the nature of the roots?
3. Can you graph the quadratic equation to visualize the roots?
4. How would the solutions change if \(b\) or \(c\) had different values?
5. What are some real-world applications of solving quadratic equations?

**Tip:** The discriminant, \(b^2 - 4ac\), can tell you whether the solutions are real or complex. If it's positive, the solutions are real!

Solve the quadratic equation 9r^2 - 5r - 10 = 0.

Learn how to solve the quadratic equation 9r^2 - 5r - 10 using the quadratic formula. This solution covers key algebraic concepts, including the use of the quadratic formula and its application to find the roots of the equation. Suitable for high school students in Grades 9-12.

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