Solve 6 / a = a / x for x, where a = (-1 + sqrt 71) / 2. Leave the answer in radical form.

Step-by-step solution

1. 1

   Name the surd and postpone substituting it. Keeping $a=\dfrac{-1+\sqrt{71}}{2}$ as a single symbol lets you do the algebra once, cleanly, and squares only one expression at the end. This is the difference between one messy calculation and three.

2. 2

   Cross-multiply the proportion. From $\dfrac{6}{a}=\dfrac{a}{x}$ (with $a,x\neq 0$):

   $6x=a^2\ \Longrightarrow\ x=\frac{a^2}{6}$

   This is the "mean proportional" relation: $a$ is the geometric mean of $6$ and $x$.

3. 3

   Square the surd with $(p+q)^2=p^2+2pq+q^2$. Squaring numerator and denominator separately:

   $a^2=\frac{\left(-1+\sqrt{71}\right)^2}{4}=\frac{1-2\sqrt{71}+71}{4}=\frac{72-2\sqrt{71}}{4}$

   Note $\left(\sqrt{71}\right)^2=71$ exactly, which is what removes one radical; the cross term $-2\sqrt{71}$ survives.

4. 4

   Reduce the fraction before dividing again. Factoring 2 out of the numerator:

   $a^2=\frac{2\left(36-\sqrt{71}\right)}{4}=\frac{36-\sqrt{71}}{2}$

5. 5

   Divide by 6 to finish.

   $x=\frac{a^2}{6}=\frac{36-\sqrt{71}}{2\cdot 6}=\frac{36-\sqrt{71}}{12}$

   Leave it exactly like this: replacing $\sqrt{71}\approx 8.426$ would give $x\approx 2.298$ and throw away the exactness the question asked for.

6. 6

   Check numerically. With $\sqrt{71}=8.42615$: $a=3.71307$ and $x=2.29782$. Then $6/a=1.61591$ and $a/x=1.61591$ — the two sides agree, confirming the surd answer.