Question

write the quadratic equation whose roots are one by root 2, 1 by root 2​

Answer 1

GIVEN :

The roots are $\frac{1}{\sqrt{2}}$ and $\frac{1}{\sqrt{2}}$

TO WRITE:

The quadratic equation with the given roots.

SOLUTION :

From the given roots  $\frac{1}{\sqrt{2}}$  and  $\frac{1}{\sqrt{2}}$

Let $\alpha=\frac{1}{\sqrt{2}}$   and $\beta=\frac{1}{\sqrt{2}}$ be the roots given.

If $\alpha$ and $\beta$ be the two roots  are given , then the formula for the quadratic equation is given by:

$x^2 - (\alpha +\beta ) x + \alpha \beta = 0$.  

We can also write a quadratic equation as

$x^2-(sum of the roots)x+product of the roots=0$

Sum of the roots is $\alpha+\beta=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}$  

$=2(\frac{1}{\sqrt{2}})$

$=\sqrt{2}$

∴ Sum of the roots is $\alpha+\beta=\sqrt{2}$

Product of the roots is $\alpha \beta=\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}$

$=\frac{1}{2}$

∴ Product of the roots is $\alpha \beta=\frac{1}{2}$

Now substitute the values in the formula we get

$x^2-(\sqrt{2})x+\frac{1}{2}=0$

∴ the quadratic equation for the given roots is $x^2-(\sqrt{2})x+\frac{1}{2}=0$