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The given equation is:

\[
12x^2 + 11x = -2
\]

To solve this equation graphically, we can first rewrite it as:

\[
12x^2 + 11x + 2 = 0
\]

This is a quadratic equation, and solving it graphically means plotting the quadratic function \( f(x) = 12x^2 + 11x + 2 \) and finding its roots (where the graph crosses the x-axis).

Now, let's solve it algebraically using the quadratic formula. The quadratic formula is:

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

For the equation \( 12x^2 + 11x + 2 = 0 \), we have:

- \( a = 12 \)
- \( b = 11 \)
- \( c = 2 \)

Plugging these values into the quadratic formula:

\[
x = \frac{-11 \pm \sqrt{11^2 - 4(12)(2)}}{2(12)}
\]

\[
x = \frac{-11 \pm \sqrt{121 - 96}}{24}
\]

\[
x = \frac{-11 \pm \sqrt{25}}{24}
\]

\[
x = \frac{-11 \pm 5}{24}
\]

Now, solving for the two possible values of \(x\):

1. \( x = \frac{-11 + 5}{24} = \frac{-6}{24} = -0.25 \)
2. \( x = \frac{-11 - 5}{24} = \frac{-16}{24} = -\frac{2}{3} \approx -0.667 \)

Thus, the solutions are approximately \( x = -0.25 \) and \( x = -0.667 \).

So the correct answer is:

\[
\boxed{x = -0.667, x = -0.25}
\]

Do you want any further details or explanations?

Here are 5 relative questions for practice:
1. What is the significance of the discriminant in the quadratic formula?
2. How can you determine the number of real solutions for a quadratic equation without solving it?
3. How does the sign of the coefficient \(a\) in a quadratic function affect its graph?
4. What are the properties of the vertex in a parabola?
5. How would factoring help in solving quadratic equations?

**Tip:** When solving quadratics, always check if the equation can be factored easily before using the quadratic formula. It can save time!

Solve 12x^2 + 11x = -2 by graphing the expressions on both sides of the equation.

Learn how to solve the quadratic equation 12x^2 + 11x + 2 = 0 by using the quadratic formula. This problem covers essential algebra concepts and graphing techniques, suitable for high school students in Grades 9-11.

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