https://file.aiquickdraw.com/mathbot/file/1725775282134rp7ogug2.jpg

The equation in the image appears to be:

\[
2q^3 + 3q^2 - 3q + 12 = 0
\]

This is a cubic equation in the variable \( q \). To solve this equation, we would typically try to find one real root, possibly using methods like factoring, trial and error, or numerical approaches, and then apply synthetic division or the quadratic formula to solve the remaining quadratic equation.

Let's attempt to solve this equation step by step.

### Step 1: Try to find rational roots using the Rational Root Theorem.

The Rational Root Theorem suggests that any rational root, if it exists, must be a factor of the constant term (12) divided by a factor of the leading coefficient (2). So, the possible rational roots could be:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{6}{2}
\]

### Step 2: Evaluate the equation for possible rational roots.

Let me test those values and find the solution.It appears that none of the possible rational roots provide a solution to the equation \( 2q^3 + 3q^2 - 3q + 12 = 0 \), as none of the evaluations yield zero.

Since the Rational Root Theorem didn't work, we may need to resort to numerical methods (like Newton's method) or use a solver to approximate the real roots.

Let me compute the approximate real root(s) of this cubic equation next.The cubic equation \( 2q^3 + 3q^2 - 3q + 12 = 0 \) has one real root and two complex roots:

- Real root: \( q \approx -2.80 \)
- Complex roots: \( q \approx 0.65 + 1.31i \) and \( q \approx 0.65 - 1.31i \)

Would you like a more detailed explanation of how the cubic equation is solved or any other clarifications?

Here are some related questions you may find helpful:
1. How is the Rational Root Theorem applied to other equations?
2. What are the different methods for solving cubic equations?
3. Can complex roots appear in all types of polynomial equations?
4. How can Newton's method be used to find real roots?
5. What is synthetic division, and how can it simplify cubic equations?

**Tip:** When solving higher-degree polynomials, graphical methods or numerical solvers are often effective in finding real roots when analytical methods are complex or fail.

Solve the cubic equation: 2q^3 + 3q^2 - 3q + 12 = 0

Learn how to solve the cubic equation 2q^3 + 3q^2 - 3q + 12 = 0 with step-by-step methods. This problem explores algebraic concepts such as the Rational Root Theorem and complex numbers. Suitable for students in Grades 10-12.

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