Question

How could this Rational Equation be solved? The answer is supposed to be 4 times the negative or positive square root of 30, and I've tried it a bunch of times, but keep getting a similar, but slightly off answer. Maybe I'm using the wrong LCD?

3/x^2-4 + 1/2 = 4/x-2

Answer 1

`(3)/(x^2-4) + (1)/(2)= (4)/(x-2)\\\\(3)/((x-2)(x+2)) + (1)/(2) = (4)/(x-2)\\\\(3)/((x-2)(x+2))= (4)/(x-2) - (1)/(2)\\\\(3)/((x-2)(x+2)) = (4(x+2))/((x-2)(x+2))-(1(x-2)(x+2))/(2(x-2)(x+2))\\\\(3)/((x-2)(x+2)) = (4x+8)/((x-2)(x+2))-(x^2-4)/(2(x-2)(x+2))\\\\(3)/((x-2)(x+2)) = (4x+8-x^2+4)/((x-2)(x+2))\\\\(x-2)(x+2)*(3)/((x-2)(x+2)) = (4x+8-x^2+4)/((x-2)(x+2))* (x-2)(x+2)\\\\3 = 4x + 8 -x^2+4`


`3 = 4x + 8 -x^2+4\\\\3 = 4x+ 12 - x^2\\\\3 = -x^2 + 4x + 12\\\\-x^2 + 4x + 9 = 0\\\\x_(1,2) = (-b \pm √(b^2-4ac))/(2a)\\\\x_(1,2) = ( -(-4) \pm √( (-4)^2 - 4 \cdot 1 \cdot (-9)) )/( 2 \cdot 1 )\\\\x_(1,2) = ( 4 \pm √( 52 ) )/( 2 )\\\\x_1 = ( 4~+~√( 52 ) )/( 2 ) = 2+√(13)\\\\x_2 = ( 4~-~√( 52 ) )/( 2 ) = 2-√(13)`

So obviously the answer they gave you is wrong. :P