Find the roots of the following quadratic equations by the factorisation method.

$3x^2+5\sqrt{5}x-10=0$

Find the roots of the following quadratic equations by the factorisation method.

$21x^2-2x+\frac{1}{21}=0$

Find whether the following equations have real roots. If real root exist, Find them

$8x^2+2x-3=0$

If $a \mathrm{and} b$ are the roots then, find $16k$ if $a^2+b^2=k$.

Find whether the given equation have real roots. If real root exist, Find them

$-2x^2+3x+2=0$

If $a \mathrm{and} b$ are the roots of the equation then find $4k$ if $a^2+b^2=k$

Find whether the following equations have real roots. If real root exist, Find them
$5x^2-2x-10=0$

If one of the root is $\frac{1-\sqrt{k}}{5}$, find $k$

Find whether the following equations have real roots. If real root exist, Find them.

$\frac{1}{2x-3}+\frac{1}{x-5}=1, x\ne \frac{3}{2}, 5$

If one of the root is $4+6\sqrt{k}$, then find $k$?

Find whether the following equations have real roots. If real root exist, Find them

$x^2+5\sqrt{5}x-70=0$

If one of the root is $2\sqrt{k}$, find $k$

Find a natural number whose square diminished by $84$ is equal to thrice of $8$ more than the given number.

A natural number, when increased by $12$, equals $160$ times its reciprocal. Find the number.