Question

${\displaystyle 25x^2+30x+7=0}$

Answer 1

Given: 

${\displaystyle 25x^2+30x+7=0}$ 

On comparing it with ${\displaystyle a{x}^2+bx+x=0}$ 

a = 25,b = 30 and c = 7
Discriminant D is given by: 

${\displaystyle D=(b^2-4ac)}$ 

=${\displaystyle {30}^2-4\times 25\times 7}$ 

=${\displaystyle 900-700}$ 

=${\displaystyle 200}$ 

=${\displaystyle 200}$ 

=${\displaystyle 200>0}$ 

Hence, the roots of the equation are real.
Roots α and β are given by: 

${\displaystyle \alpha =\frac{-b+\sqrt{D}}{2a}=\frac{-30+\sqrt{200}}{2\times 25}=\frac{-30+10\sqrt{20}}{50}=\frac{10(-3+\sqrt{2})}{50}=\frac{(-3+\sqrt{2})}{5}}$ 

${\displaystyle \beta =\frac{-b-\sqrt{D}}{2a}=\frac{-30-\sqrt{200}}{2\times 25}=\frac{-30-10\sqrt{20}}{50}=\frac{10(-3-\sqrt{2})}{50}=\frac{(-3-\sqrt{2})}{5}}$ 

Thus, the roots of the equation are ${\displaystyle \frac{(-3+\sqrt{2})}{5}}$ and ${\displaystyle \frac{(-3-\sqrt{2})}{5}}$