MathematicsChapter 2: Quadratic Equations

The Quadratic Formula

For ax^2 + bx + c = 0, x = (-b +/- sqrt(b^2 - 4ac)) / (2a).

Learning objectives

Explain the quadratic formula using the correct definitions, conditions, and notation.
Apply x = -b +/- sqrt(D)2a, D = b^2 - 4ac 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 = 1, b = -5, c = 6

D = (-5)² - 4(1)(6)

D = 25 - 24

D = 1

Where / symbols

THE QUADRATIC FORMULA: For ax^2 + bx + c = 0, x = (-b +/- sqrt(b^2 - 4ac)) / (2a). The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square.
THE FORMULA: Numerator is -b, not b. Denominator is 2a, not a. The +/- gives two roots (unless D=0). Keep the fraction bar over the whole numerator.
EXAMPLE NUMBERS: x^2 - 5x + 6 = 0: a=1, b=-5, c=6. -b=5. D=25-24=1. x = (5 +/- 1)/2 so x=3 or x=2. Same as factoring.

Relationship

x = -b +/- sqrt(D)2a
D = b² - 4ac

Overview

Plug in, then simplify:

The Formula

Numerator is -b, not b. Denominator is 2a, not a. The +/- gives two roots (unless D=0). Keep the fraction bar over the whole numerator.

Examples a, b, c from standard form · sqrt only of D · Two values if D>0

Example Numbers

x² - 5x + 6 = 0: a=1, b=-5, c=6. -b=5. D=25-24=1. x = 5 +/- 12 so x=3 or x=2. Same as factoring.

Examples D=1 · x=(5+1)/2=3 · x=(5-1)/2=2

WHAT — THE QUADRATIC FORMULA

For ax² + bx + c = 0, x = -b +/- sqrt(b² - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

The Quadratic Formula 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 THE FORMULA, then EXAMPLE NUMBERS. Worked: solve x^2 - 5x + 6 = 0 by formula. a = 1, b = -5, c = 6 D = (-5)^2 - 4(1)(6) D = 25 - 24 D = 1 x = 5 +/- 12 x = 62 = 3 or x = 42 = 2 Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

solve x² - 5x + 6 = 0 by formula.
a = 1, b = -5, c = 6
D = (-5)² - 4(1)(6)
D = 25 - 24
D = 1
x = 5 +/- 12
x = 62 = 3 or x = 42 = 2

In simple words

(-5)^2 is 25, not -25. Square the whole of b, including its sign.

Key points

x = -b +/- sqrt(D)2a
D = b² - 4ac
-b and 2a
matches factoring

Exam preparation

Exam Focus

Quick revision

Revision Zone

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