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Write down the x-coordinates of the points where $y = x^2 - x$ and $y = x + 1$ cross. - OCR - GCSE Maths - Question 22 - 2020 - Paper 3

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Write down the x-coordinates of the points where $y = x^2 - x$ and $y = x + 1$ cross.

Worked Solution & Example Answer:Write down the x-coordinates of the points where $y = x^2 - x$ and $y = x + 1$ cross. - OCR - GCSE Maths - Question 22 - 2020 - Paper 3

Step 1

Set the equations equal to each other

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Answer

To find the x-coordinates where the two functions intersect, set them equal to each other:

x2−x=x+1x^2 - x = x + 1

Step 2

Rearrange the equation

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Answer

Rearranging gives:

x2−2x−1=0x^2 - 2x - 1 = 0

Step 3

Use the quadratic formula

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Answer

To solve for x, use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a=1a = 1, b=−2b = -2, and c=−1c = -1.

Calculate the discriminant:

b2−4ac=(−2)2−4(1)(−1)=4+4=8b^2 - 4ac = (-2)^2 - 4(1)(-1) = 4 + 4 = 8

Step 4

Find the roots

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Answer

Now plugging into the quadratic formula gives:

x=2±82=1±2x = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}

Thus, the x-coordinates are:

x=1+2x = 1 + \sqrt{2}

and

x=1−2x = 1 - \sqrt{2}

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