To solve the equation \( \frac{3}{2}x^2 + 11 = 59 \), follow these steps:

1. **Subtract 11 from both sides** to isolate the quadratic term:
   \[
   \frac{3}{2}x^2 + 11 - 11 = 59 - 11
   \]
   \[
   \frac{3}{2}x^2 = 48
   \]

2. **Multiply both sides by 2/3** to solve for \( x^2 \):
   \[
   x^2 = \frac{48 \times 2}{3} = \frac{96}{3} = 32
   \]

3. **Take the square root** of both sides to solve for \( x \):
   \[
   x = \pm\sqrt{32}
   \]

4. **Simplify the square root**:
   \[
   x = \pm\sqrt{32} = \pm\sqrt{16 \times 2} = \pm 4\sqrt{2}
   \]

So, the solutions are:
\[
x = 4\sqrt{2} \quad \text{or} \quad x = -4\sqrt{2}
\]

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

Here are some related questions you might find interesting:

1. How do you solve quadratic equations by completing the square?
2. What are the steps for solving quadratic equations using the quadratic formula?
3. How do you factor quadratic equations?
4. How do you simplify square roots?
5. What are the properties of square roots in algebra?

**Tip:** When solving quadratic equations, always check if the equation can be factored easily before using other methods.

Explore the step-by-step solution to the quadratic equation 3/2x^2 + 11 = 59. This guide covers key algebraic concepts such as isolating the quadratic term and simplifying square roots. Perfect for students in Grades 8-10.

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