Question - Determining the Correct Equation that Expresses a Variable in Terms of Others
Solution:
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.
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