Question

Quadratic equation whose roots are the number $\frac{1}{10-\sqrt{72}}$ and $\frac{1}{10+6\sqrt{2}}$ is $a{x}^2+bx+c=0$ the the value of $a+b+c$ is ?

Answer 1

Given,

Roots, $\frac{1}{10-\sqrt{72}},\frac{1}{10+6\sqrt{2}}$

$\Rightarrow \frac{1}{10-6\sqrt{2}},\frac{1}{10+6\sqrt{2}}$

sum of roots $=\frac{1}{10-6\sqrt{2}}+\frac{1}{10-6\sqrt{2}}=\frac{20}{28}$

product of roots $=\frac{1}{10-6\sqrt{2}}\times \frac{1}{10-6\sqrt{2}}=\frac{1}{28}$

$p\left(x\right)=x^2-(sum-of-roots)x+(product-of-roots)$

$x^2-\left(\frac{20}{28}\right)x+\left(\frac{1}{28}\right)=0$

$28x^2-20x+1=0$

$a{x}^2+bx+c=0\Rightarrow a=28,b=-20,c=1$

$a+b+c=28-20+1=9$