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# Help asap! By tomorrow please!

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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
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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!

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

$$a\cdot b=5$$

$$b=\frac{5}{a}$$

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

$$2a^3-a^2-25=0$$

$$a=2.5\\ b=2\\ a^2+b^2=10.25$$

!

Oct 28, 2020
#2
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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