Understanding the Equation a/b b/a 2: A Comprehensive Guide
IntroductionIntroduction
Exploring algebraic equations and solving them is a fundamental part of understanding mathematical principles. In this article, we will delve into an intriguing problem: If a/b b/a 2, what is the value of ab?Solving the Equation
Given the problem statement, we start by defining the equation as such:[ frac{a}{b} times frac{b}{a} 2 ]
Since (frac{a}{b} times frac{b}{a} 1), the given equation simplifies to:[ 1 2 ]
This is a contradiction, so let's define a new variable to simplify the process. Let (x frac{a}{b}). The equation then becomes:[ x times frac{1}{x} 2 ]
Simplifying further:[ x^2 - 1 2x ]
Rearranging terms gives us the quadratic equation:[ x^2 - 2x - 1 0 ]
Using the quadratic formula (x frac{-b pm sqrt{b^2 - 4ac}}{2a}), we get:[ x frac{2 pm sqrt{4 4}}{2} 1 pm sqrt{2} ]
Since (x frac{a}{b}), we have (frac{a}{b} 1 sqrt{2}) or (frac{a}{b} 1 - sqrt{2}). However, the negative value doesn't fit the original problem's context since (frac{a}{b} times frac{b}{a}) should equal 1, and thus (frac{a}{b} 1 sqrt{2}) is the feasible solution. Consequently, (a b(1 sqrt{2})). Substituting (a b(1 sqrt{2})) into the expression for (ab):[ ab b times b(1 sqrt{2}) b^2(1 sqrt{2}) ]
Thus, we find that ab (1 sqrt{2})b^2, where (b) is any non-zero value.