+14995 Suppose [(1/a)+(1/b)] * (a-b) equals to 1. And a*b=5. What is a^2+b^2?
Hello Guest!
\(\color{BrickRed} (\frac{1}{a}+\frac{1}{b})(a-b)=1\\ \frac{a}{a}-\frac{b}{a}+\frac{a}{b}-\frac{b}{b}=1\\ \color{blue}\frac{a}{b}-\frac{b}{a}=1\)
\(a\cdot b=5\)
\(b=\frac{5}{a}\)
\(\frac{2a}{5}-\frac{5}{a^2}=1\\ \frac{2a^3-25}{a^2}=1\\ 2a^3-25=a^2 \)
\(2a^3-a^2-25=0\)
\(a=2.5\\ b=2\\ a^2+b^2=10.25\)
!
+421 This problem is a common example of clever manipulations.
The question: $$\frac{1}{a} + \frac{1}{b} \cdot (a-b) = 1 \text{ and } a \cdot b = 5. \text { What is } a^2 + b^2 ? $$
The solution:
This can be simplified to $(a^2-b^2) = ab$, but we know that $ab=5,$ so $a^2-b^2=5.$ We now know that by substitution, $$a^2+b^2=2a^2+5.$$
Be careful asinus, I think you made a mistake!
$a^2+b^2=5\sqrt{5}!$
