2(a + b) = (3 + \sqrt5)(a - b) - Decision Point
Mastering the Equation: Solving 2(a + b) = (3 + √5)(a - b)
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.
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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
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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
