The roots of q quadratic $x^{2} - \sqrt{20 x} + 2 = 0$ $x^2-sqrt(20x)+2=0$ are c and d. Without using calculator show that $\frac{1}{c} + \frac{1}{d} = \sqrt{5}$ $1/c+1/d=sqrt(5)$ ?

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$\frac{1}{c} + \frac{1}{d} = \sqrt{5}$ $1/c+1/d=sqrt(5)$

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Answer 1 · 2018-01-22T09:43:08

See the proof below

If the roots of a quadratic equation $a x^{2} + b x + c = 0$ $ax^2+bx+c=0$ are

$α$ $alpha$ and $β$ $beta$ then,

$α + β = - \frac{b}{a}$ $alpha+beta=-b/a$

and

$α β = \frac{c}{a}$ $alpha beta=c/a$

Here the quadratic equation is $x^{2} - \sqrt{20} x + 2 = 0$ $x^2-sqrt20 x+2=0$

and the roots are $c$ $c$ and $d$ $d$

Therefore,

$c + d = \sqrt{20}$ $c+d=sqrt20$

$c d = 2$ $cd=2$

so,

$\frac{1}{c} + \frac{1}{d} = \frac{d + c}{c d}$ $1/c+1/d=(d+c)/(cd)$

$= \frac{\sqrt{20}}{2}$ $=(sqrt20)/2$

$= \frac{2 \sqrt{5}}{2}$ $=(2sqrt5)/2$

$= \sqrt{5}$ $=sqrt5$

$Q E D$ $QED$