Example Question - mathematical rearrangement

Determining the Correct Equation that Expresses a Variable in Terms of Others

Given: \( c = \frac{a^2 + b^2}{ab} \) Need to express \( a \) in terms of \( b \) and \( c \). Multiply both sides by \( ab \): \( abc = a^2 + b^2 \) Rearranging to express \( a \): \( a^2 - abc + b^2 = 0 \) Factor by grouping: \( a^2 - ab(c - 1) - b^2 = 0 \) Let \( a = b \) and \( (c - 1) = –a \): \( a^2 - a(b) - b^2 = 0 \) Hence, \( a = \frac{b \pm \sqrt{b^2 - 4(1)(-b^2)}}{2(1)} \) Simplify under the square root: \( a = \frac{b \pm \sqrt{5b^2}}{2} \) Since \( a \) and \( b \) are distinct positive numbers, use the positive root: \( a = \frac{b + \sqrt{5}b}{2} \) Factor out \( b \): \( a = \frac{b(1 + \sqrt{5})}{2} \) Therefore, the correct equation that expresses \( a \) in terms of \( b \) and \( c \) is: \( a = \frac{b(1 + \sqrt{5})}{2} \) Which corresponds to answer choice A.