To solve the quadratic equation $7x^2 + 11x = 6$, we follow these steps:

Step 1: Set the equation to zero

First, move all terms to one side of the equation: $7x^2 + 11x - 6 = 0$

Step 2: Apply the quadratic formula

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ In this case, $a = 7$, $b = 11$, and $c = -6$.

Step 3: Calculate the discriminant

$\text{Discriminant} = b^2 - 4ac = 11^2 - 4(7)(-6)$ $\text{Discriminant} = 121 + 168 = 289$

Step 4: Solve for $x$

Now, substitute the values into the quadratic formula: $x = \frac{-11 \pm \sqrt{289}}{2(7)}$ $x = \frac{-11 \pm 17}{14}$

This gives us two possible solutions for $x$:

1. $x_1 = \frac{-11 + 17}{14} = \frac{6}{14} = \frac{3}{7}$
2. $x_2 = \frac{-11 - 17}{14} = \frac{-28}{14} = -2$

Final Answer

The solutions are: $x = \frac{3}{7} \quad \text{or} \quad x = -2$

Would you like more details or have any questions?

Here are five related questions:

1. How would you solve a quadratic equation by factoring?
2. Can you derive the quadratic formula from completing the square?
3. What does the discriminant tell us about the nature of the roots?
4. How do you solve quadratic equations using the method of completing the square?
5. What are the applications of quadratic equations in real-world problems?

Tip: When using the quadratic formula, always check the discriminant first to determine the nature of the roots (real and distinct, real and repeated, or complex).