Mastering the Equation: Solving 2(a + b) = (3 + √5)(a - b)

Understanding algebraic equations is a fundamental skill in mathematics, and equations involving radicals like √5 often appear in advanced algebra, trigonometry, and mathematical modeling. One such equation that learners frequently encounter is:

> 2(a + b) = (3 + √5)(a - b)

Understanding the Context

Whether you’re solving for one variable in terms of the other or exploring deeper algebraic properties, mastering this equation strengthens your problem-solving abilities. In this article, we’ll guide you step-by-step through simplifying, solving, and interpreting the equation — all optimized for clarity and SEO-friendly content.

What Does the Equation Represent?

The equation
2(a + b) = (3 + √5)(a − b)
is a linear relationship linking two expressions involving variables a and b. The term (3 + √5) is an irrational coefficient, making this equation ideal for practicing simplification and algebraic manipulation, especially when working with radicals.

2(a + b) = (3 + \sqrt{5})(a - b)

Key Insights

Step-by-Step Solution

Step 1: Expand both sides

Start by expanding both sides to eliminate parentheses:

Left-hand side:
2(a + b) = 2a + 2b

Continue Reading

New total volume = \( 50 + x \) liters.
New water content = \( 30 + x \) liters.
Set up the equation: \( \frac{30 + x}{50 + x} = 0.8 \).

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Final Thoughts

Right-hand side:
(3 + √5)(a − b) = 3a − 3b + a√5 − b√5

So, the equation becomes:
2a + 2b = 3a − 3b + a√5 − b√5

Step 2: Move all terms to one side

Collect every term to the left to group like terms:

2a + 2b − 3a + 3b − a√5 + b√5 = 0

Combine like terms:

  • a-terms: 2a − 3a = −a
  • b-terms: 2b + 3b = 5b
  • radical terms: −a√5 + b√5 = √5(b − a)

Resulting equation:
−a + 5b + √5(b − a) = 0

Step 3: Factor intelligent grouping