Question

if x=
$\sqrt{5} - \sqrt{3} \div \sqrt{5} + \sqrt{3}$
and y=
$\sqrt{5} + \sqrt{3} \div \sqrt{5} - \sqrt{3}$
then find
x^2+y^2-6xy​

Answer 1

Answer:

56

Step-by-step explanation:

Rationalize the denominator with identity.

The identity is $a^2-b^2=(a+b)(a-b)$.

We need to find x.

Simply multiply $\sqrt{5} -\sqrt{3}$ on denominator and numerator.

$x=\frac{(\sqrt{5} -\sqrt{3} )^2}{(\sqrt{5} +\sqrt{3})(\sqrt{5} -\sqrt{3} ) } =\frac{5-2\sqrt{15} +3}{2} =4-\sqrt{15}$

We need to find y.

Simply multiply $\sqrt{5} +\sqrt{3}$ on denominator and numerator.

$y=\frac{(\sqrt{5} +\sqrt{3} )^2}{(\sqrt{5} +\sqrt{3})(\sqrt{5} -\sqrt{3} ) } =\frac{5+2\sqrt{15} +3}{2} =4+\sqrt{15}$

We know that $(x+y)^2=x^2+2xy+y^2$.

Simply subtract 8xy from both sides, and we get this.

• $(x+y)^2-8xy=x^2+y^2-6xy$

We know that $x+y=8$ and $xy=1$.

Here is the answer : $8^2-8=56$