Learn8300Papers 1–3 · Foundation tierAlgebraQuadratic equations

Algebra · Subtopic 4 of 9

Quadratic Equations

Rearrange to $ax^2+bx+c=0$ . If it factorises, each bracket gives a solution. Otherwise use the quadratic formula, or complete the square, which also gives the turning point of the graph. In context, reject solutions that make no sense.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Solve quadratics by factorising, including when $a\ne1$ .
Use the quadratic formula, giving answers to a stated accuracy.
Complete the square and find the turning point.
Form a quadratic from a context and interpret its solutions.

Pause and think

If $A\times B=0$ , what can you say about $A$ and $B$ ? Why does that help solve $(x-2)(x+5)=0$ ?

Section 2 of 14

Core idea

Start hereA garden of 40 m²

A rectangular garden is 3 m longer than it is wide and has area 40 m². How wide? Writing width $x$ gives $x(x+3)=40$ , an equation with an $x^2$ in it: a quadratic.

The big idea

Rearrange to $ax^2+bx+c=0$ . Factorise if possible; otherwise complete the square or use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .

1. Factorising

y = x² − 2x − 3 crosses the x-axis at −1 and 3

x^2-2x-3=(x+1)(x-3)

. Each bracket equals zero at one root.

Worked example $x^2-2x-3=0$

Solve.

Goal: Roots.
Strategy: If two numbers multiply to 0, one of them is 0.

$(x+1)(x-3)=0$
Multiply to $-3$ , add to $-2$ .
$x=-1\text{ or }x=3$
Each bracket zero.

Answer

-1

3

✓ Check

9-6-3=0

✓.

2. Completing the square and the formula

Worked example $x^2+6x+2=0$

Solve exactly and to 2 d.p.

Goal: Roots.
Strategy: $(x+3)^2=x^2+6x+9$ .

$(x+3)^2-9+2=0$
Complete the square.
$(x+3)^2=7\Rightarrow x=-3\pm\sqrt7$
Root both sides (±).
$x=-0.35\text{ or }-5.65$
$\sqrt7=2.646$ .

Answer

-3\pm\sqrt7

✓ CheckSum

-6=-b

✓.

The formula

$x^2+\tfrac bax+\tfrac ca=0$
Divide by $a$ .
$\left(x+\tfrac b{2a}\right)^2=\frac{b^2-4ac}{4a^2}$
Complete the square.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Root and rearrange.

Worked example $2x^2-5x-1=0$

Solve to 2 d.p.

Goal: Roots.
Strategy: Formula, careful with signs.

$a=2,\ b=-5,\ c=-1$
Identify.
$x=\frac{5\pm\sqrt{25+8}}{4}$
$-b=5$ ; $-4ac=+8$ .
$x=2.69\text{ or }-0.19$
$\sqrt{33}=5.745$ .

Answer

2.69

-0.19

✓ CheckProduct

=\frac ca=-0.5

2.69\times(-0.19)\approx-0.5

✓.

Worked exampleThe garden

Width $x$ , length $x+3$ , area 40.

Goal: Dimensions.
Strategy: Form and solve the quadratic.

$x^2+3x-40=0$
Expand and rearrange.
$(x+8)(x-5)=0$
Multiply to $-40$ , add to $3$ .
$x=5$
Width can't be $-8$ .

Answer5 m by 8 m.

✓ Check

5\times8=40

✓.

Common misconceptionDividing by x

x^2=5x

: factorise

x(x-5)=0

to keep

x=0

Summary

Rearrange to $=0$ .
Factorise, complete the square, or formula.

Root: A solution.

History4000-year-old quadratics. Babylonian clay tablets from about

1800

BC solve problems like "a square's area minus its side is

870

" with a step-by-step recipe that is essentially completing the square. They had no symbols at all: every step was written in words.

Fun factThe discriminant gossips. The part under the square root,

b^2-4ac

, tells you how many solutions there are before you work them out: positive means two, zero means one, negative means none. It discriminates between the cases, hence the name.

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