Question

find the quadratic equation,if one of the roots is√5-√3​

Answer 1

Quadratic equation if one of the roots is$\sqrt{5} - \sqrt{3}$ is

$x^{2} -2\sqrt{5} x+2$

Step-by-step explanation:

As $\sqrt{5} - \sqrt{3}$ is irrational root, therefore the other root is $\sqrt{5} + \sqrt{3}$

As we know the sum of the roots is = $-\frac{b}{a}$

And the products of the roots is = $\frac{c}{a}$

Therefore, the sum of the roots is $\sqrt{5} - \sqrt{3}$ +$\sqrt{5} + \sqrt{3}$ = $2\sqrt{5}$

Therefore, the product of the roots is( $\sqrt{5} - \sqrt{3}$)($\sqrt{5} + \sqrt{3}$) = 5-3 = 2

Thus equating we get

$-\frac{b}{a}$= $2\sqrt{5}$

and

$\frac{c}{a}$= 2

Now a  quadratic equation is in the form $ax^{2} +bx+c$

Therefore putting the values we get,

$x^{2} -2\sqrt{5} x+2$

Quadratic equation if one of the roots is$\sqrt{5} - \sqrt{3}$ is

$x^{2} -2\sqrt{5} x+2$

Answer 2

The quadratic equation which has √5-√3​ is x²-2√5x+2=0

The roots are conjugate if roots are irrational

one of the root is √5-√3​ hence the second root has to be √5+√3

general equation of quadratic is ax²+bx+c=0

sum of roots = ​-b/a

=2√5

product of roots=c/a

=2

hence the equation is x²-2√5x+2=0