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 Context

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

Key Insights

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

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To find the minimum value of \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \), given \( x + y + z = 6 \), use the AM-HM inequality:
\frac{x + y + z}{3} \geq \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}
Substitute \( x + y + z = 6 \):

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Final Thoughts

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