Solve equations using completing square method.
Ax^2  + bx + c = 0
12r^2  – 17r – 5 = 0

To solve the quadratic equation \( 12r^2 - 17r - 5 = 0 \) using the method of completing the square, follow these steps:

1. **Rewrite the equation** in the form \( ar^2 + br + c = 0 \):
   \[
   12r^2 - 17r - 5 = 0
   \]

2. **Divide through by the coefficient of \( r^2 \) (if it is not 1)**:
   \[
   r^2 - \frac{17}{12}r - \frac{5}{12} = 0
   \]

3. **Move the constant term to the other side**:
   \[
   r^2 - \frac{17}{12}r = \frac{5}{12}
   \]

4. **Complete the square**:
   - Take half of the coefficient of \( r \), which is \( -\frac{17}{12} \), and square it:
     \[
     \left(\frac{-\frac{17}{12}}{2}\right)^2 = \left(\frac{-17}{24}\right)^2 = \frac{289}{576}
     \]
   - Add this square to both sides of the equation:
     \[
     r^2 - \frac{17}{12}r + \frac{289}{576} = \frac{5}{12} + \frac{289}{576}
     \]

5. **Simplify the equation**:
   \[
   \frac{5}{12} = \frac{240}{576}
   \]
   \[
   \frac{240}{576} + \frac{289}{576} = \frac{529}{576}
   \]
   \[
   r^2 - \frac{17}{12}r + \frac{289}{576} = \frac{529}{576}
   \]

6. **Express the left side as a perfect square**:
   \[
   \left(r - \frac{17}{24}\right)^2 = \frac{529}{576}
   \]

7. **Take the square root of both sides**:
   \[
   r - \frac{17}{24} = \pm \sqrt{\frac{529}{576}}
   \]
   \[
   r - \frac{17}{24} = \pm \frac{23}{24}
   \]

8. **Solve for \( r \)**:
   \[
   r = \frac{17}{24} \pm \frac{23}{24}
   \]

   This gives us two solutions:
   \[
   r = \frac{17 + 23}{24} = \frac{40}{24} = \frac{5}{3}
   \]
   \[
   r = \frac{17 - 23}{24} = \frac{-6}{24} = -\frac{1}{4}
   \]

So the solutions to the quadratic equation \( 12r^2 - 17r - 5 = 0 \) are:
\[
r = \frac{5}{3} \quad \text{and} \quad r = -\frac{1}{4}
\]

Would you like more details on any specific step or have any questions?

Here are some related questions to expand your understanding:

1. How can you verify the solutions to a quadratic equation?
2. What are the advantages of completing the square compared to other methods?
3. Can the completing square method be used for all quadratic equations?
4. How do you derive the quadratic formula using the completing square method?
5. How does the discriminant relate to the solutions found by completing the square?
6. What are some real-world applications of solving quadratic equations?
7. How do you handle completing the square for equations with a coefficient other than 1 for \( r^2 \)?
8. What are the key differences between factoring and completing the square methods?

**Tip**: Always check your final solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the quadratic equation 12r^2 - 17r - 5 = 0 using the completing the square method with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations and completing the square. Suitable for students in Grades 10-12.

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