The given equation is:

$6x^2 - 11x - 7 = 0$

This is a quadratic equation in standard form $ax^2 + bx + c = 0$, where:

• $a = 6$
• $b = -11$
• $c = -7$

To solve this, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step 1: Identify the coefficients

Here, $a = 6$, $b = -11$, and $c = -7$.

Step 2: Calculate the discriminant

The discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac$

Substituting the values of $a$, $b$, and $c$:

$\Delta = (-11)^2 - 4(6)(-7)$ $\Delta = 121 + 168$ $\Delta = 289$

Step 3: Apply the quadratic formula

Now, substitute $\Delta = 289$ into the quadratic formula:

$x = \frac{-(-11) \pm \sqrt{289}}{2(6)}$ $x = \frac{11 \pm 17}{12}$

Step 4: Find the two solutions

1. For the positive root:

$x_1 = \frac{11 + 17}{12} = \frac{28}{12} = \frac{7}{3}$

2. For the negative root:

$x_2 = \frac{11 - 17}{12} = \frac{-6}{12} = -\frac{1}{2}$

Final Answer:

The solutions to the equation $6x^2 - 11x - 7 = 0$ are:

$x = \frac{7}{3} \quad \text{or} \quad x = -\frac{1}{2}$

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Would you like further explanation on any of the steps or any questions on this topic?

Here are 5 related questions:

1. What happens if the discriminant $\Delta$ is negative?
2. How can you solve quadratic equations by factoring?
3. What is the graphical interpretation of a quadratic equation and its roots?
4. How does the quadratic formula relate to the vertex form of a parabola?
5. Can you solve a quadratic equation using completing the square method?

Tip: When using the quadratic formula, always double-check the values you substitute for $a$, $b$, and $c$ to avoid calculation mistakes!