To solve the equation \( r^2 - 16r + 11 = -9 \), follow these steps:

1. **Move all terms to one side of the equation**:  
   \[
   r^2 - 16r + 11 + 9 = 0
   \]
   Simplify this to:
   \[
   r^2 - 16r + 20 = 0
   \]

2. **Solve the quadratic equation** using the quadratic formula:
   \[
   r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For the equation \( r^2 - 16r + 20 = 0 \), the coefficients are:
   \[
   a = 1, \quad b = -16, \quad c = 20
   \]

3. **Substitute the coefficients** into the quadratic formula:
   \[
   r = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(20)}}{2(1)}
   \]
   Simplifying further:
   \[
   r = \frac{16 \pm \sqrt{256 - 80}}{2}
   \]
   \[
   r = \frac{16 \pm \sqrt{176}}{2}
   \]
   \[
   r = \frac{16 \pm \sqrt{16 \times 11}}{2}
   \]
   \[
   r = \frac{16 \pm 4\sqrt{11}}{2}
   \]

4. **Simplify the expression**:
   \[
   r = 8 \pm 2\sqrt{11}
   \]

So, the solutions are:
\[
r = 8 + 2\sqrt{11} \quad \text{or} \quad r = 8 - 2\sqrt{11}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do you derive the quadratic formula?
2. Can you solve a quadratic equation by completing the square?
3. What are the roots of the equation \( r^2 - 16r + 20 = 0 \) when factored?
4. How can you verify the solutions to a quadratic equation?
5. What does the discriminant tell you about the nature of the roots?

**Tip:** Always double-check your discriminant value to ensure that you haven't made an arithmetic error in the quadratic formula!

Learn how to solve the quadratic equation r^2 - 16r + 11 = -9 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic equation and the quadratic formula theorem. Suitable for students in Grades 8-10.

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