https://file.aiquickdraw.com/mathbot/file/1728896367704pj608n22.jpg

To solve the equation \( 6x^2 - 5x = 56 \) using the zero product property, follow these steps:

### Step 1: Set the equation to zero
First, subtract 56 from both sides of the equation to set it equal to zero:
\[
6x^2 - 5x - 56 = 0
\]

### Step 2: Factor the quadratic equation
We need to factor \( 6x^2 - 5x - 56 = 0 \). We are looking for two numbers that multiply to \( 6 \times (-56) = -336 \) and add up to \(-5\) (the coefficient of \(x\)).

Those two numbers are \( -21 \) and \( 16 \), because:
\[
-21 \times 16 = -336 \quad \text{and} \quad -21 + 16 = -5.
\]

Next, decompose the middle term:
\[
6x^2 - 21x + 16x - 56 = 0
\]

### Step 3: Group the terms and factor by grouping
Group the terms:
\[
(6x^2 - 21x) + (16x - 56) = 0
\]

Factor each group:
\[
3x(2x - 7) + 8(2x - 7) = 0
\]

Now, factor out the common binomial \( (2x - 7) \):
\[
(3x + 8)(2x - 7) = 0
\]

### Step 4: Solve for \(x\)
Using the zero product property, set each factor equal to zero:
1. \( 3x + 8 = 0 \quad \Rightarrow \quad x = -\frac{8}{3} \)
2. \( 2x - 7 = 0 \quad \Rightarrow \quad x = \frac{7}{2} \)

### Final answer:
The solutions are \( x = -\frac{8}{3} \) or \( x = \frac{7}{2} \), which corresponds to the second option in the image.

Would you like more details on any part of this solution?

### Related Questions:
1. How do you factor quadratic equations?
2. What is the zero product property in algebra?
3. How do you solve quadratic equations by grouping?
4. What other methods can you use to solve quadratic equations?
5. Can the quadratic formula solve this equation? How?

### Tip:
Always check if a quadratic equation can be factored before using the quadratic formula—it often simplifies the solution process.

Use the zero product property to find the solutions to the equation 6x^2 - 5x = 56.

Learn how to solve the quadratic equation 6x^2 - 5x = 56 using the zero product property and factoring. Follow a detailed step-by-step solution suitable for students in Grades 8-10.

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