solve the equation by factoring: 12x^2 -11x =15

To solve the equation \( 12x^2 - 11x = 15 \) by factoring, follow these steps:

1. **Set the equation to zero:**  
   Subtract 15 from both sides of the equation:  
   \[
   12x^2 - 11x - 15 = 0
   \]

2. **Factor the quadratic equation:**  
   We need to find two numbers that multiply to \( 12 \times (-15) = -180 \) and add to \( -11 \).  
   The numbers that work are \( -20 \) and \( 9 \) because:
   \[
   -20 \times 9 = -180 \quad \text{and} \quad -20 + 9 = -11
   \]
   
3. **Rewrite the middle term using these numbers:**  
   Break the middle term, \(-11x\), into \(-20x + 9x\):
   \[
   12x^2 - 20x + 9x - 15 = 0
   \]

4. **Factor by grouping:**  
   Group the terms into two pairs:
   \[
   (12x^2 - 20x) + (9x - 15) = 0
   \]
   Factor out the common factors in each group:
   \[
   4x(3x - 5) + 3(3x - 5) = 0
   \]

5. **Factor out the common binomial factor:**  
   Notice that \( (3x - 5) \) is a common factor:
   \[
   (4x + 3)(3x - 5) = 0
   \]

6. **Set each factor to zero and solve for \( x \):**  
   \[
   4x + 3 = 0 \quad \text{or} \quad 3x - 5 = 0
   \]
   
   - For \( 4x + 3 = 0 \):
   \[
   4x = -3
   \]
   \[
   x = -\frac{3}{4}
   \]

   - For \( 3x - 5 = 0 \):
   \[
   3x = 5
   \]
   \[
   x = \frac{5}{3}
   \]

The solutions to the equation \( 12x^2 - 11x - 15 = 0 \) are:
\[
x = -\frac{3}{4} \quad \text{and} \quad x = \frac{5}{3}
\]

Would you like more details or have any questions?

### Relative Questions:
1. How do you factor a quadratic equation when the leading coefficient is not 1?
2. What is the significance of the numbers found for factoring by grouping?
3. How do you check if the solutions to a quadratic equation are correct?
4. Can the quadratic formula be used to solve this equation instead?
5. How does factoring by grouping work in other types of polynomial equations?

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

Learn how to solve the quadratic equation 12x^2 - 11x = 15 by factoring. This step-by-step solution covers key algebraic concepts such as factoring, the quadratic formula, and the zero product property. Suitable for students in Grades 9-10.

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