Solve (x/4)-(2x/x+2) = 1. Give your solutions to 2 decimal places. You must show your working.

We will begin by eliminating fractions in the equation. To do this, the equation will be multiplied by 4 and (x+2), as these are the denominators of the two fractions.
Multiplying by 4 gives:x - (8x/x+2) = 4Multiplying by (x+2) gives:x(x+2) - 8x = 4(x+2)Expanding gives:x^2 + 2x - 8x = 4x + 8Equating to zero and then collecting like terms gives:x^2 + 2x - 8x - 4x - 8 = 0x^2 - 10x - 8 = 0
As seen above, we end up with a quadratic equation. However, this equation does not have integer solutions, hence factorisation by observation is tricky. In a case such as this one, the Quadratic Formula may be used. The Quadratic Formula is given below.
For a given quadratic equation of form ax^2 + bx + c = 0x = ( -b ± squareroot( b^2 - 4ac ) ) / 2aNote the "±" sign and that this implies 2 solutions for x; one for the case of "+ squareroot" and one for "- squareroot".
Hence for our equation, x^2 - 10x - 8 = 0, a = 1, b = -10 and c = -8. Applying the Quadratic Formula,x = ( --10 ± squareroot( (-10)^2 - 41*-8 ) ) / 2*1x = ( 10 ± squareroot(132) ) / 2
Therefore, the two exact solutions are,x = 5 + squareroot(33) and x = 5 - squareroot(33)for the cases of "+ squareroot" and "- squareroot" respectively. Giving answers to 2 decimal places, as required,
x = 10.74 and x = -0.74 (correct to 2 decimal places)