MathematicsChapter 2: Quadratic Equations

Quadratic Equations

A quadratic equation can be written ax^2 + bx + c = 0 with a not 0.

Learning objectives

Explain quadratic equations using the correct definitions, conditions, and notation.
Apply ax^2 + bx + c = 0, a != 0 correctly in routine and exam-style questions.
Show each important step, simplify the result, and check that it satisfies the original conditions.

Definition

Mathematical form

a = 2, b = 0, c = -8

c = 0 with a not 0. The highest power of x i

a = 2

b = 0

Where / symbols

QUADRATIC EQUATIONS: A quadratic equation can be written ax^2 + bx + c = 0 with a not 0. The highest power of x is 2. 3x+2=0 is linear, not quadratic. Collect every term on one side so the other side is 0 before you name a, b, c.
WHAT COUNTS: x^2 - 4 = 0 is quadratic (b=0). 5x^2 + x = 0 is quadratic (c=0). 2x^2 = 8 is quadratic after 2x^2 - 8 = 0. 3x+2=0 is not.
COEFFICIENTS: In 3x^2 - 7x + 2 = 0, a=3, b=-7, c=2. The sign is part of b. Leading coefficient is a, not the last number.

Relationship

ax² + bx + c = 0
a != 0

Overview

Recognise it, then name a, b, c:

What Counts

x^2 - 4 = 0 is quadratic (b=0). 5x^2 + x = 0 is quadratic (c=0). 2x^2 = 8 is quadratic after 2x^2 - 8 = 0. 3x+2=0 is not.

Examples x^2-4=0 · 2x^2=8 · 3x+2=0 is linear

Coefficients

In 3x^2 - 7x + 2 = 0, a=3, b=-7, c=2. The sign is part of b. Leading coefficient is a, not the last number.

Examples a=3, b=-7, c=2 · x^2+6x+9: a=1,b=6,c=9 · Do not swap a and c

WHAT — QUADRATIC EQUATIONS

A quadratic equation can be written ax² + bx + c = 0 with a not 0. The highest power of x is 2. 3x+2=0 is linear, not quadratic. Collect every term on one side so the other side is 0 before you name a, b, c. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

Quadratic Equations matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through WHAT COUNTS, then COEFFICIENTS. Worked: write 2x^2 = 8 in standard form and name a, b, c. 2x^2 = 8 2x^2 - 8 = 0 a = 2, b = 0, c = -8 Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

write 2x^2 = 8 in standard form and name a, b, c.
2x² = 8
2x² - 8 = 0
a = 2, b = 0, c = -8

In simple words

If a = 0 the equation is no longer quadratic. The x^2 term must actually be there.

Key points

ax² + bx + c = 0
a != 0
degree 2
a is x^2

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.