Question

find a and b: $\sqrt{5}-2/\sqrt{5} +2-\sqrt{5} +2/\sqrt{5} -2=a+b\sqrt{5}$

Answer 1

Answer:

Given:

$\dfrac{\sqrt5 - 2}{\sqrt5 + 2} - \dfrac{\sqrt5 + 2}{\sqrt5 - 2} = a + b\sqrt5$

Solution:

$\implies \dfrac{\sqrt5 -2}{ \sqrt5 + 2} \times \dfrac{\sqrt5 - 2}{\sqrt5 -2} \\ - \dfrac{\sqrt5 +2}{\sqrt5 -2} \times \dfrac{\sqrt5 + 2}{\sqrt5 + 2} = a+ \sqrt5b$

$\implies \dfrac{(\sqrt5 - 2)^2}{ (\sqrt5)^2 - 2^2} - \dfrac{(\sqrt5 + 2)^2}{(\sqrt5)^2 - 2^2}$

$\implies \dfrac{9 - 4\sqrt5}{5 - 4} - \dfrac{9 + 4\sqrt5}{5-4} = a + \sqrt5 b$

$\implies 9 - 4 \sqrt5 - 9 - 4\sqrt5 = a+ \sqrt5 b$

$\implies -8 \sqrt5 = a + \sqrt5 b$

$\boxed{\implies{\boxed{ a = 0, \ b = -8}}}$

Answer 2

a=0 and b= -8

formulas to be used

(a+b)^2= a^2+b^2+2ab

(a-b)^2= a^2+b^2-2ab

(a^2-b^2)= (a-b)(a+b)

check the above pic for solution