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Quadratic Equations (H)

Quadratic Equations (H)

Crack higher-tier quadratics: solve equations by factorising, completing the square or the quadratic formula, and read graphs to find roots, turning points and line intersections.

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Fascinating Fact:

In biomechanics, if s(t) = 3t² − 12t + 5, the minimum occurs at t = −b ÷ (2a) = 12 ÷ 6 = 2. Substituting gives s(2) = −7, the lowest point.

In GCSE Maths, quadratic equations model curved graphs and many real situations. You’ll solve ax² + bx + c = 0 by factorising, completing the square, or using the quadratic formula, and interpret graphs to find key features.

Key Terms

Quadratic equation: An equation of the form ax² + bx + c = 0 with a ≠ 0.
Vertex (turning point): The maximum or minimum point of a parabola; it occurs at x = −b/(2a).
Discriminant: The value b² − 4ac telling you the number of real solutions (two, one, or none).

Frequently Asked Questions (Click to see answers)

How do I solve a quadratic by factorising?

Write it as (x + p)(x + q) = 0. Then set each factor to zero: x = −p or x = −q. This works when the quadratic factorises neatly.

What is the quadratic formula and when should I use it?

Use x = (−b ± √(b² − 4ac)) / (2a). It always solves ax² + bx + c = 0, even when factorising is hard.

How do I complete the square to solve or find the vertex?

For x² + bx + c, write (x + b/2)² − (b/2)² + c. Read the vertex from (x + b/2)² + k, or solve by setting the square equal to a number.

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You can find more about this topic by visiting BBC Bitesize - Solving quadratic equations

Author: Sally Thompson (MSc Operational Research, Secondary Maths Teacher & Quiz Writer)

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