The given expression is a quadratic equation:  
\[ 7x^2 - 32x - 60 = 0 \]

Let's solve this using the **quadratic formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where:
- \( a = 7 \)
- \( b = -32 \)
- \( c = -60 \)

### Step 1: Calculate the discriminant (\( \Delta \)):
\[
\Delta = b^2 - 4ac = (-32)^2 - 4(7)(-60) = 1024 + 1680 = 2704
\]

### Step 2: Find the square root of the discriminant:
\[
\sqrt{\Delta} = \sqrt{2704} = 52
\]

### Step 3: Apply the quadratic formula:
\[
x = \frac{-(-32) \pm 52}{2(7)} = \frac{32 \pm 52}{14}
\]

### Step 4: Calculate the two possible solutions:
1. \( x_1 = \frac{32 + 52}{14} = \frac{84}{14} = 6 \)
2. \( x_2 = \frac{32 - 52}{14} = \frac{-20}{14} = -\frac{10}{7} \)

### Final Answer:
The solutions are:
\[
x = 6 \quad \text{or} \quad x = -\frac{10}{7}
\]

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

### Related Questions:
1. How do you derive the quadratic formula?
2. What does the discriminant tell us about the roots of a quadratic equation?
3. Can you explain how the discriminant changes the nature of the solutions (real vs complex)?
4. How can you verify the solutions of a quadratic equation?
5. What is factoring, and how would it apply if this equation were factorable?

**Tip**: When solving quadratic equations, always check the discriminant first. It tells you whether the roots are real or complex.

Learn how to solve the quadratic equation 7x^2 - 32x - 60 using the quadratic formula. This step-by-step guide covers key algebraic concepts such as the discriminant and quadratic formula. Suitable for students in Grades 9-10.

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