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IGCSE & GCSE • Mathematics (0580 / 4MA1)10 min read

Quadratic Equations & Functions

Methods for solving quadratics, discriminant analysis, and graph properties

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Must-Know Formulas for this Topic
Quadratic Formula
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
for ax2+bx+c=0\text{for } ax^2 + bx + c = 0
The Discriminant
Δ=b2−4ac\Delta = b^2 - 4ac
>0  ⟹  2 roots;=0  ⟹  1;<0  ⟹  0> 0 \implies 2\text{ roots}; = 0 \implies 1; < 0 \implies 0
Line of Symmetry
x=−b2ax = -\frac{b}{2a}
Vertex x-coordinate\text{Vertex x-coordinate}

1. Three Key Ways to Solve Quadratics

Factorization: Write as (px + q)(rx + s) = 0. Best when factors are integers and easily recognizable.

Quadratic Formula: Universal method that works for any quadratic equation, particularly when questions state 'give answers to 2 decimal places' or 'in surd form'.

Completing the Square: Write in the vertex form a(x + p)² + q = 0. Essential for locating turning points and sketch curves.

ax2+bx+c=a(x+b2a)2+(c−b24a)ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

2. The Discriminant and Nature of Roots

If b² - 4ac > 0: Two distinct real roots (graph cuts x-axis at two points).

If b² - 4ac = 0: One repeated real root (graph touches x-axis, tangent to axis).

If b² - 4ac < 0: No real roots (graph does not touch or intersect the x-axis).

Examiner Pitfall Alert
  • When taking the square root in the formula, always include the ± sign.
  • Check whether the question asks for exact values (surd form) or rounded decimals.
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