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Algebra 01 (H)

Tackle higher-tier algebra with confidence. Learn completing the square, find a quadratic’s vertex, and use it to solve problems about heights, speeds, and maximum or minimum values.

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Fascinating Fact:

An arch follows y = −0.2x² + 3x + 5. Complete the square to get the vertex. That gives the maximum headroom for tall vehicles safely.

In GCSE Maths (Higher Tier), you use completing the square to rewrite quadratics and locate the vertex. This reveals maximum or minimum values and helps model real contexts like arches and trajectories.

Key Terms

Quadratic: An expression or equation with x² as the highest power, such as ax² + bx + c.
Vertex: The turning point of a parabola, giving the quadratic’s maximum or minimum value.
Completing the square: Rewriting ax²+bx+c in the form a(x + p)² + q to identify the vertex and solve.

Frequently Asked Questions (Click to see answers)

How do you complete the square for a quadratic?

For x² + bx + c, write x² + bx as (x + b/2)² − (b/2)², then add c. For ax² terms, factor out a first.

How do I find the vertex using completing the square?

Rewrite as a(x + p)² + q. The vertex is ( −p , q ). If a > 0 it is a minimum, and if a < 0 it is a maximum.

When should I use completing the square instead of factorising?

Use it when a quadratic does not factorise neatly, when you need the vertex, or when deriving the turning point and the equation’s maximum or minimum value.