Solving the Equation: 2(x + 1) + 3x = x(x + 1) โ€“ 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
Solved 2x-1 (x+1) | Chegg.com

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Solved 2x-1 (x+1) | Chegg.com
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Solved: |2 x+1|
Solved: the tonowimg expressions 3x(x+1)(x-1) and 2x^2(x-1)(x+3) [algebra]

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
= 2ยทx + 2ยท1 + 3x
= 2x + 2 + 3x
= (2x + 3x) + 2
= 5x + 2

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Final Thoughts

Right-hand side (RHS):
x(x + 1)
= xยทx + xยท1
= xยฒ + x

So the equation becomes:
5x + 2 = xยฒ + 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:
xยฒ + x โ€“ 5x โ€“ 2 = 0
โ†’ xยฒ โ€“ 4x โ€“ 2 = 0

Step 3: Solve the Quadratic Equation

Use the quadratic formula:
x = [โˆ’b ยฑ โˆš(bยฒ โ€“ 4ac)] / (2a)
For xยฒ โ€“ 4x โ€“ 2 = 0,
a = 1, b = โ€“4, c = โ€“2

Calculate the discriminant:
ฮ” = bยฒ โ€“ 4ac = (โˆ’4)ยฒ โ€“ 4(1)(โ€“2) = 16 + 8 = 24

Roots:
x = [4 ยฑ โˆš24] / 2
โˆš24 = โˆš(4ร—6) = 2โˆš6, so
x = [4 ยฑ 2โˆš6] / 2
โ†’ x = 2 ยฑ โˆš6