Solving quadratics and their graphs

Factorising

Put the equation in the form ax² + bx + c = 0 first. For a not 1, look for two numbers that multiply to ac and add to b, then split the middle term.

2x² - 5x - 3 = 2x² - 6x + x - 3 = (2x + 1)(x - 3).

The quadratic formula

x = (-b +/- sqrt(b² - 4ac)) / (2a). Use it when the quadratic does not factorise, or when the question asks for surds or decimals.

The graph y = ax² + bx + c

• a > 0: u-shaped; a < 0: n-shaped
• crosses the y-axis at (0, c)
• crosses the x-axis at the roots
• the line of symmetry is x = -b/(2a), halfway between the roots

Worked example

Solve 2x² - 5x - 3 = 0.

1. ac = -6. The numbers -6 and 1 multiply to -6 and add to -5.
2. (2x + 1)(x - 3) = 0
3. 2x + 1 = 0 or x - 3 = 0

Answer: x = -1/2 or x = 3

Check yourself

1. Solve x² - 7x + 12 = 0.
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   x = 3 or x = 4. (x - 3)(x - 4) = 0.

2. Solve 2x² + 5x - 3 = 0.
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   x = 1/2 or x = -3. (2x - 1)(x + 3) = 0, so x = 1/2 or x = -3.

3. Solve x² - 4x - 1 = 0. Give the positive root to 2 decimal places.
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   4.24. x = (4 +/- sqrt(16 + 4))/2 = 2 +/- sqrt(5). The positive root is 4.24.

4. Where does y = (x - 2)(x + 5) cross the y-axis? Give the y-coordinate.
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   -10. Put x = 0: (-2)(5) = -10.

5. Find the x-coordinate of the turning point of y = x² - 6x + 2.
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   3. The line of symmetry is x = -b/(2a) = 6/2 = 3.

6. Which describes the graph of y = -2x² + 3x + 1?
   • n-shaped, crossing the y-axis at (0, 1)
   • u-shaped, crossing the y-axis at (0, 1)
   • n-shaped, crossing the y-axis at (0, -2)
   • u-shaped, crossing the y-axis at (0, 3)
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   n-shaped, crossing the y-axis at (0, 1). a = -2 is negative, so the curve is n-shaped. Putting x = 0 gives y = 1.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.