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

February 22, 2026

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.

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.

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

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.

Final Answer: What Are the Solutions?

✅ The complete solution set is: x = √141 and x = −√141

✅ The smallest real solution is: −√141, since it is negative and smaller than positive √141.

💡 Pro Tip: If this equation came from modeling, the smallest root might represent a valid threshold or boundary in contexts like physics or finance.

Why This Matters – Real-World Applications

Solving such equations is essential in:

Engineering (calculating dimensions under constraints)
Economics (finding break-even points)
Science (modeling exponential growth with transformations)
Computer graphics (radius and coordinate systems)

Understanding how to isolate variables, manipulate square roots, and interpret sign solutions prepares you for advanced math like calculus and algebra-based problem solving.

Summary

Start: 3x² + 2 = 425
Isolate: x² = 141
Solve: x = ±√141
Smallest solution: x = −√141
Express related forms carefully: √141 − 1 ≠ smallest root but may appear in transformations

Mastering these steps not only solves the equation but strengthens foundational algebra skills — key for academic success and practical problem solving.

If you found this helpful, explore step-by-step guides on solving higher-degree equations, quadratic forms, or applications in science and finance — keep practicing, and math will become clearer every day!

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