Sum of Two Numbers

Two different positive numbers a and b each differ from their reciprocals by 1. What is a+b ?
Source: NCTM Mathematics Teacher

SOLUTION
We have
\displaystyle a-\frac{1}{a}=1
Simplify
a^2-1=a
a^2-a-1=0
Solve for a by completing the square
a^2-a+\displaystyle\frac{1}{4}=\frac{5}{4}
\displaystyle\left (a-\frac{1}{2}\right )^2=\frac{5}{4}
\displaystyle a-\frac{1}{2}=\pm\frac{\sqrt{5}}{2}
a=\displaystyle\frac{1\pm\sqrt{5}}{2}
=\displaystyle\frac{1+\sqrt{5}}{2}>0
Similarly
\displaystyle\frac{1}{b}-b=1
Simplify
1-b^2=b
b^2+b-1=0
Solve for b by completing the square
b^2+b+\displaystyle\frac{1}{4}=\frac{5}{4}
\displaystyle\left (b+\frac{1}{2}\right )^2=\frac{5}{4}
\displaystyle b+\frac{1}{2}=\pm\frac{\sqrt{5}}{2}
b=\displaystyle\frac{-1\pm\sqrt{5}}{2}
=\displaystyle\frac{-1+\sqrt{5}}{2}>0
a+b=\displaystyle\frac{1+\sqrt{5}}{2}+\frac{-1+\sqrt{5}}{2}
=\sqrt{5}

Answer: \sqrt{5}

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About mvtrinh

Retired high school math teacher.
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