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Suppose [(1/a)+(1/b)] * (a-b) equals to 1. And a*b=5. What is a^2+b^2?

 Oct 28, 2020
 #1
avatar+10581 
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Suppose [(1/a)+(1/b)] * (a-b) equals to 1. And a*b=5. What is a^2+b^2?

 

Hello Guest!

 

\(\color{BrickRed} (\frac{1}{a}+\frac{1}{b})(a-b)=1\\ \frac{a}{a}-\frac{b}{a}+\frac{a}{b}-\frac{b}{b}=1\\ \color{blue}\frac{a}{b}-\frac{b}{a}=1\)

\(a\cdot b=5\)

\(b=\frac{5}{a}\)

\(\frac{2a}{5}-\frac{5}{a^2}=1\\ \frac{2a^3-25}{a^2}=1\\ 2a^3-25=a^2 \)

\(2a^3-a^2-25=0\)

\(a=2.5\\ b=2\\ a^2+b^2=10.25\)

laugh  !

 Oct 28, 2020
 #2
avatar+260 
0

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!

$a^2+b^2=5\sqrt{5}!$

 Oct 29, 2020

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