To solve the equation $\frac{x}{3} \cdot x + 1 = 5$, follow these steps:

Step 1: Expand the terms

The left-hand side simplifies to: $\frac{x}{3} \cdot x = \frac{x^2}{3}$ Thus, the equation becomes: $\frac{x^2}{3} + 1 = 5$

Step 2: Subtract 1 from both sides

$\frac{x^2}{3} = 4$

Step 3: Multiply through by 3 to eliminate the fraction

$x^2 = 12$

Step 4: Solve for $x$ by taking the square root

$x = \pm \sqrt{12}$ Simplify the square root: $x = \pm 2\sqrt{3}$

Final Answer:

$x = 2\sqrt{3} \quad \text{or} \quad x = -2\sqrt{3}$

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Would you like a detailed explanation for any step or have other related questions? Here are some relevant questions for practice:

1. How do we simplify square roots like $\sqrt{12}$?
2. What are the properties of square roots when dealing with positive and negative values?
3. How can we verify solutions to quadratic equations?
4. How does multiplying through by a denominator affect an equation?
5. Can you explain how $x^2 = c$ has two solutions?

Tip: Always simplify radicals as much as possible to leave answers in their most compact form.