What Is a Quadratic Equation?

A quadratic equation is any equation where the highest power of $x$ is 2. The standard form is:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are numbers and $a \neq 0$.

Term	Role	Example ($3x^2 - 5x + 2 = 0$)
$a$	Coefficient of $x^2$	$a = 3$
$b$	Coefficient of $x$	$b = -5$
$c$	Constant term	$c = 2$

Is It Quadratic? Three Quick Tests

Equation	Quadratic?	Why?
$5x^2 + 2x = 0$	Yes	Highest power is $x^2$, $a = 5 \neq 0$
$x^3 - 4x + 1 = 0$	No	Highest power is $x^3$ (cubic)
$x(x - 3) = x^2 + 5$	No	Expand: $x^2 - 3x = x^2 + 5 \Rightarrow -3x - 5 = 0$ (the $x^2$ cancels — linear!)

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Step 1: Get Into Standard Form

Before you can solve, rearrange everything so the right side equals zero.

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Step 2: Factorise the Quadratic

Factorisation turns $ax^2 + bx + c$ into a product of two brackets. This is the skill O-Level examiners test most often.

When $a = 1$: The Factor-Pair Method

For $x^2 + bx + c$, find two numbers that:

• Multiply to give $c$
• Add to give $b$

Sign Rules Cheat Sheet

$c$ (constant)	$b$ (middle term)	Signs in brackets
Positive $(+)$	Positive $(+)$	Both positive: $(x + \_)(x + \_)$
Positive $(+)$	Negative $(-)$	Both negative: $(x - \_)(x - \_)$
Negative $(-)$	Positive $(+)$	Bigger is positive: $(x + \_)(x - \_)$
Negative $(-)$	Negative $(-)$	Bigger is negative: $(x - \_)(x + \_)$

Special Case: Difference of Two Squares

When there’s no middle term ($b = 0$) and $c$ is negative:

$x^2 - 49 = (x + 7)(x - 7)$

The formula is: $a^2 - b^2 = (a + b)(a - b)$

When $a \neq 1$: The Cross Method

For expressions like $2x^2 + 5x + 3$, the factor-pair method gets trickier. Use the cross method (also called the “cross-multiplication” or “AC method”).

Use the interactive tool below to see the cross method in action:

Quadratic Factorizer

Enter coefficients to factor ax² + bx + c into (px + r)(qx + s)

Try an example:

x² + 5x + 6x² − 7x + 122x² + 7x + 33x² − 10x + 86x² + 11x − 10

x²

+

x

+

FactorizeReset

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Step 3: Solve Using the Zero Product Property

Once you’ve factorised, solving is easy thanks to this rule:

If $A \times B = 0$, then either $A = 0$ or $B = 0$ (or both).

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Step 4: The Simpler Case — $x^2 = k$

Some quadratics don’t need factorisation at all. If you can isolate $x^2$, just take the square root.

What if $x^2 = -16$? No real solution — you can’t square a real number and get a negative result. The graph of $y = x^2 + 16$ never touches the $x$-axis.

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Step 5: Read the Graph (Parabola Basics)

Every quadratic equation $y = ax^2 + bx + c$ produces a parabola when graphed. Here’s what to look for:

Shape: Is It Smiling or Frowning?

Value of $a$	Shape	Description
$a > 0$	U-shape (smile)	Opens upward — has a minimum point
$a < 0$	∩-shape (frown)	Opens downward — has a maximum point

Key Features of a Parabola

Feature	What It Is	How to Find It
$y$-intercept	Where the curve crosses the $y$-axis	Set $x = 0$: it’s just $c$
$x$-intercepts (roots)	Where the curve crosses the $x$-axis	Solve $ax^2 + bx + c = 0$
Vertex	The turning point (min or max)	$x = -\frac{b}{2a}$, then find $y$
Axis of symmetry	The vertical line through the vertex	$x = -\frac{b}{2a}$

How Many Roots?

The graph tells you how many solutions the equation $ax^2 + bx + c = 0$ has:

Graph Crosses $x$-axis…	Number of Real Roots	Discriminant ($b^2 - 4ac$)
Twice	2 distinct roots	$> 0$
Once (touches)	1 repeated root	$= 0$
Never	0 real roots	$< 0$

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The 5 Most Common Quadratic Mistakes

#	Mistake	Example	Fix
1	Forgetting $\pm$ with square roots	Writing $x = 3$ instead of $x = \pm 3$	Always write $\pm$ when taking $\sqrt{}$
2	Dividing by $x$ instead of factorising	Losing the $x = 0$ solution	Factor out $x$ first, then use Zero Product Property
3	Wrong signs in factor pairs	$(x + 3)(x + 5)$ instead of $(x - 3)(x - 5)$	Use the sign rules cheat sheet above
4	Not rearranging to standard form	Trying to factorise $x^2 + 3x = 10$ directly	Move everything to one side: $x^2 + 3x - 10 = 0$
5	Forgetting to check solutions	Accepting answers that don’t satisfy the original equation	Substitute each answer back into the original equation

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Complete Worked Example: Putting It All Together

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Quick Reference: The Complete Method

1. Rearrange to standard form: $ax^2 + bx + c = 0$
2. Factorise the expression (factor pairs, cross method, or difference of squares)
3. Apply Zero Product Property: set each bracket $= 0$
4. Solve each mini-equation for $x$
5. Check by substituting back into the original equation
6. Reject solutions that don’t make sense in context (e.g., negative lengths)

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