Solve \[ \frac{1}{2x - 1} + \frac{3}{x - 1} = 1 \] Give your answer in the form \( p \pm \sqrt{q} \over 2 \) where p and q are integers. - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 1

By multiplying both sides by the common denominator, we eliminate the fractions:

[ (x - 1) + 3(2x - 1) = (2x - 1)(x - 1) ]

Expanding this gives:

[ x - 1 + 6x - 3 = 2x^2 - 3x + 1 ]

Combining like terms results in:

[ 7x - 4 = 2x^2 - 3x + 1 ]

Rearranging results in a standard quadratic equation:

[ 2x^2 - 10x + 5 = 0 ]

To find the values of ( x ), apply the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) where ( a = 2, b = -10, c = 5 ):

[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2} ]

Calculating the discriminant:

[ 100 - 40 = 60 ]

Thus, we can write:

[ x = \frac{10 \pm \sqrt{60}}{4} = \frac{10 \pm 2\sqrt{15}}{4} = \frac{5 \pm \sqrt{15}}{2} ]