Solving Quadratics
Solve quadratic equations by factorising and applying the null factor law, finding the values of x that make the equation equal to zero.
Quadratic Equations
A quadratic equation has the form ax2 + bx + c = 0. A quadratic can have two solutions, one solution (a repeated root), or no real solutions. The solutions are called the roots of the equation.
Two solutions
The parabola crosses the x-axis at two points.
One solution
The parabola just touches the x-axis (repeated root).
No real solutions
The parabola does not touch the x-axis.
The Null Factor Law
The null factor law states: if the product of two expressions equals zero, then at least one of those expressions must equal zero. This is the key principle for solving factorised quadratics.
Null Factor Law
If A × B = 0, then A = 0 or B = 0 (or both).
How to Apply It
Example: Solve (x − 3)(x + 5) = 0
Step 1: Apply the null factor law: x − 3 = 0 or x + 5 = 0
Step 2: Solve each: x = 3 or x = −5
Solutions: x = 3 or x = −5
Solving by Factorising
To solve a quadratic by factorising: rearrange to standard form (= 0), factorise the left-hand side, then apply the null factor law.
Important: You must always rearrange so one side equals zero before factorising. Never cancel or divide both sides by a variable expression containing x.
Step 1
Rearrange to ax2 + bx + c = 0
Step 2
Factorise the left side
Step 3
Apply the null factor law
Key Vocabulary
Quadratic equation
An equation of the form ax2 + bx + c = 0 where a ≠ 0.
Null factor law
If A × B = 0, then A = 0 or B = 0.
Roots / solutions
The values of x that satisfy the equation; where the parabola crosses the x-axis.
Repeated root
When a quadratic has only one solution; the parabola touches but does not cross the x-axis.
Worked Examples
Solve x2 − 5x + 6 = 0
Step 1: Already in standard form.
Step 2: Factorise: find two numbers that multiply to 6 and add to −5. Use −2 and −3.
(x − 2)(x − 3) = 0
Step 3: Apply null factor law: x − 2 = 0 or x − 3 = 0
Solutions: x = 2 or x = 3
Solve x2 = 3x (rearranging first)
Step 1: Rearrange: x2 − 3x = 0
Step 2: Factorise by taking out common factor: x(x − 3) = 0
Step 3: Null factor law: x = 0 or x − 3 = 0
Solutions: x = 0 or x = 3
Solve x2 + 6x + 9 = 0 (repeated root)
Step 1: Already in standard form.
Step 2: Recognise as a perfect square: (x + 3)2 = 0
Step 3: x + 3 = 0
Solution: x = −3 (repeated root)
Knowledge Check
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Key Concepts Summary
- ●Always rearrange a quadratic equation to standard form (ax2 + bx + c = 0) before solving.
- ●Factorise the left side, then apply the null factor law: if A × B = 0, then A = 0 or B = 0.
- ●A quadratic can have two solutions, one repeated solution, or no real solutions.
- ●Never divide both sides by a variable — you risk losing a solution (often x = 0).
- ●Always check your solutions by substituting back into the original equation.