2(x+1) + 3x = x(x+1)

February 22, 2026

Solving the Equation: 2(x + 1) + 3x = x(x + 1) – A Comprehensive Guide

When faced with algebraic equations, simplifying both sides and solving for the variable can sometimes seem challenging. One common equation that students and math enthusiasts often explore is: 2(x + 1) + 3x = x(x + 1)

This equation represents a powerful exercise in algebraic manipulation and problem-solving. In this SEO-optimized article, we break down the full process of solving 2(x + 1) + 3x = x(x + 1), providing clear steps, real-world applications, and useful keywords to help improve search visibility for educators and learners alike.

Understanding the Equation

The equation 2(x + 1) + 3x = x(x + 1) contains both linear and quadratic components. The left side involves distribution and combination of like terms, while the right side expands into a polynomial. Solving this equation helps reinforce skills in:

Distributing parentheses
Combining like terms
Expanding binomials
Solving first-degree and quadratic equations

Step-by-Step Solution

Step 1: Expand Both Sides

Start by expanding each term using algebraic properties.

Left-hand side (LHS): 2(x + 1) + 3x = 2·x + 2·1 + 3x = 2x + 2 + 3x = (2x + 3x) + 2 = 5x + 2

Right-hand side (RHS): x(x + 1) = x·x + x·1 = x² + x

So the equation becomes: 5x + 2 = x² + x

Step 2: Move All Terms to One Side

To solve for x, bring all terms to one side to form a standard quadratic equation: x² + x – 5x – 2 = 0 → x² – 4x – 2 = 0

Step 3: Solve the Quadratic Equation

Use the quadratic formula: x = [−b ± √(b² – 4ac)] / (2a) For x² – 4x – 2 = 0, a = 1, b = –4, c = –2

Calculate the discriminant: Δ = b² – 4ac = (−4)² – 4(1)(–2) = 16 + 8 = 24

Roots: x = [4 ± √24] / 2 √24 = √(4×6) = 2√6, so x = [4 ± 2√6] / 2 → x = 2 ± √6

Final Answer

The solutions are: x = 2 + √6 and x = 2 − √6

These are real, irrational roots confirming a valid quadratic solution.

Why This Equation Matters (Applications)

This type of equation models real-world phenomena such as:

Financial break-even analysis, where costs and revenue interact nonlinearly
Physics problems involving motion with variable acceleration
Optimization scenarios in economics and engineering

Mastering its solution strengthens algebraic intuition critical for STEM fields.

Tips for Mastering Similar Problems

Practice factoring and expanding – fluency reduces errors.
Always simplify fully before solving.
Double-check each step—small mistakes change outcomes.
Use online algebra tools or graphing calculators to verify, but manual solving ensures deeper understanding.

Key SEO Keywords to Target

Optimize your content with these high-impact keywords and phrases:

Solve 2(x + 1) + 3x = x(x + 1)
Step-by-step quadratic equation solution
Solve rational and irrational roots
Algebra 2 equation solving techniques
First-degree and second-degree equations
Learn to balance algebra terms
Real-world applications of quadratic equations

Conclusion

The equation 2(x + 1) + 3x = x(x + 1) is a prime example of how combining linear and quadratic expressions challenges students to apply fundamental algebraic principles. By following structured steps—expanding, simplifying, rearranging, and applying the quadratic formula—learners gain not just answers, but powerful analytical skills. Whether for homework, exams, or personal growth, mastering such equations opens doors in mathematics and beyond.

Keywords included for SEO: “solve 2(x+1) + 3x = x(x+1)”, “quadratic equation steps”, “algebraic equation solving”, “learn quadratic formula”, “step-by-step equation solving”, “x = 2 + √6 and x = 2 − √6”, “algebraic manipulation techniques”