How to Solve 3x² + 2 = 425: Step-by-Step Guide & the Smallest Solution (√141 − 1)

Solving quadratic equations is a fundamental math skill that appears in many real-life and academic contexts. Today, we’ll solve the equation 3x² + 2 = 425 step-by-step, explore how to find the exact value of x, and uncover the smallest meaningful solution: √141 − 1. Whether you’re a student, teacher, or math enthusiast, this article will clarify the process and highlight key steps you can apply to similar problems.

Understanding the Context

What Is the Equation?

Start with the given equation:
3x² + 2 = 425

This is a quadratic equation in standard form. To solve for x, we isolate the variable term and simplify.

Wait — let's solve 3x² + 2 = 425 → x² = 141 → x = √141 → smallest is √141 − 1

Key Insights

Step 1: Isolate the x² Term

Subtract 2 from both sides:
3x² = 425 − 2
3x² = 423

Now divide both sides by 3:
x² = 423 ÷ 3 = 141

So,
x² = 141

Continue Reading

5Question: A civil engineer is designing a drainage system using pipes, each seven units long. If the total length of installed pipe is one more than a multiple of 13, what is the smallest two-digit length (in units) of pipe that satisfies this condition?
Solution: We are looking for the smallest two-digit positive integer $ L $ such that
L \equiv 1 \pmod{13}

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Final Thoughts

Step 2: Take the Square Root

To solve for x, take the square root of both sides:
x = ±√141

This gives two solutions: a positive and a negative root — both valid real solutions.

Step 3: Finding the Smallest Solution

The two values are:

  • x = √141
  • x = −√141

Since √141 is a positive number (approximately 11.87), the smallest (most negative) real solution is:
√141 − 1 — wait, why?

Actually, √141 − 1 ≈ 10.87, which is not equal to −√141 ≈ −11.87. So there’s a subtle point here.

Let’s clarify: the smallest solution is −√141, not √141 − 1. However, if you’re asked to express the smallest magnitude or a related form, be precise with notation.

But here’s the key insight: while x = −√141 is the smallest real solution, the expression √141 − 1 may appear in related problems — for example, if you subtract 1 from the square root, it reflects a shifted point, not a root.