GCSE Edexcel Maths The Circle: Solve \[ \frac{1}{2x

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

Question 20

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

Solve \[ \frac{1}{2x - 1} + \frac{3}{x - 1} = 1 \] Give your answer in the form $ p \pm \frac{\sqrt{q}}{2} $ where $ p $ and $ q $ are integers.

Worked Solution & Example Answer:Solve \[ \frac{1}{2x - 1} + \frac{3}{x - 1} = 1 \] Give your answer in the form $ p \pm \frac{\sqrt{q}}{2} $ where $ p $ and $ q $ are integers. - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 1

Step 1

Step 1: Combine the fractions

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Answer

To solve the equation, start by finding a common denominator for the left side, which is ( (2x - 1)(x - 1) ).

Rewriting the equation gives: [ \frac{(x - 1) + 3(2x - 1)}{(2x - 1)(x - 1)} = 1 ]

Step 2

Step 2: Clear the denominator

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Answer

Next, eliminate the denominator by multiplying both sides by ( (2x - 1)(x - 1) ): [ (x - 1) + 3(2x - 1) = (2x - 1)(x - 1) ] Expanding both sides results in: [ x - 1 + 6x - 3 = 2x^2 - 3x + 1 ] Which simplifies to: [ 7x - 4 = 2x^2 - 3x + 1 ]

Step 3

Step 3: Rearrange into standard form

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Answer

Reorganizing the equation gives: [ 2x^2 - 10x + 5 = 0 ]

Step 4

Step 4: Use the quadratic formula

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Answer

Applying the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) where ( a = 2, b = -10, c = 5 ): [ \Delta = (-10)^2 - 4 \times 2 \times 5 = 100 - 40 = 60] Thus: [ x = \frac{10 \pm \sqrt{60}}{4} = \frac{10 \pm 2\sqrt{15}}{4} = \frac{5 \pm \sqrt{15}}{2}] So, in the required form, ( p = 5 ) and ( q = 15 ).

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

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

Step 1: Combine the fractions

Step 2: Clear the denominator

Step 3: Rearrange into standard form

Step 4: Use the quadratic formula

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