Solve this
Question:

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

Solution:

Given:

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

On comparing it with $a x^{2}+b x+c=0$, we get:

$a=25, b=30$ and $c=7$

Discriminant $D$ is given by :

$D=\left(b^{2}-4 a c\right)$

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

$=900-700$

$=200$

$=200>0$

Hence, the roots of the equation are real.

Roots $\alpha$ and $\beta$ are given by :

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

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

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

 

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