Quadratic Inequalities

Solving quadratic inequalities: between the roots for < 0, outside for > 0, rearranging first, and inequalities from a shape's area (Higher).

In the exam for Edexcel IGCSE Maths 4MA1, Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 23 real past-paper questions below

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Where Is the Curve Below Zero? (Higher)

A quadratic inequality asks where a curve is above or below the x-axis. When the $x^{2}$ term is positive, the curve is a U shape.

Rule

Solve it as “= 0” first. For a U, the two roots split the line into three parts: the curve is below 0 between them and above 0 outside them.

Key idea

Always sketch the curve. It stops you picking the wrong part.

Common mistake

Using this rule when $x^{2}$ has a minus in front. Then the curve is upside down (∩) and the rule turns round. Multiply everything by −1 first: the Negative x² card shows how.

Less Than: Between the Roots

For “< 0” or “≤ 0” the answer is one inequality between the roots.

Worked example

1
$2 y^{2} - 7 y - 30 \leq 0$ — the question
2
$(2 y + 5) (y - 6) = 0 \Rightarrow y = - 2.5, 6$ — the roots
3
$- 2.5 \leq y \leq 6$ — below 0: between the roots, keep ≤

Greater Than: Outside the Roots

For “> 0” or “≥ 0” the answer has two parts: left of the smaller root, or right of the bigger one.

Worked example

1
$4 x^{2} - 5 x - 6 > 0$ — The question. It already has 0 on the right, so find the roots.
2
$(4 x + 3) (x - 2) = 0$ — Factorise and set it equal to 0. Check: −8x + 3x = −5x, and 3 × (−2) = −6.
3
$x = - \frac{3}{4} or x = 2$ — The roots: 4x + 3 = 0 gives x = −¾, and x − 2 = 0 gives x = 2.
4
$x < - \frac{3}{4} or x > 2$ — The U is above 0 outside the roots. That is two pieces, so join them with “or”.

Common mistake

Writing 2 < x < −¾. No number is both above 2 and below −¾. Outside answers need two separate inequalities.

A trap: $x^{2} > 9$ . It looks easy, but it has two parts too.

Worked example

1
$x^{2} > 9$ — The question. Treat it like any other quadratic inequality.
2
$x^{2} - 9 > 0$ — Take 9 from both sides, so 0 is on the right.
3
$(x + 3) (x - 3) > 0$ — Factorise. It is a difference of two squares, because 9 = 3².
4
$x = - 3 or x = 3$ — The roots: each bracket equals 0.
5
$x < - 3 or x > 3$ — Greater than 0, so outside the roots. Check x = −4: (−4)² = 16, and 16 > 9.

Common mistake

Writing x > ±3, or only x > 3. Square-rooting loses the negative side, but x = −4 works too.

Number Line and Whole Numbers

Some questions ask you to show the answer on a number line, or to list the whole numbers (integers) that work.

Rule

Open circle for < or >, filled circle for ≤ or ≥. Between the roots is one shaded piece. Outside is two arrows pointing away from each other.

List the integers that satisfy $2 x^{2} - 5 x - 12 < 0$ .

Worked example

1
$2 x^{2} - 5 x - 12 < 0$ — The question. Find the roots first.
2
$(2 x + 3) (x - 4) < 0$ — Factorise. Check: −8x + 3x = −5x, and 3 × (−4) = −12.
3
$x = - 1.5 or x = 4$ — The roots: 2x + 3 = 0 gives x = −1.5, and x − 4 = 0 gives x = 4.
4
$- 1.5 < x < 4$ — Less than 0, so between the roots. Both circles are open: −1.5 and 4 are not included.
5
$- 1, 0, 1, 2, 3$ — The whole numbers above −1.5 and below 4. 4 is out because of <. Do not forget 0.

Common mistake

Listing −2. It is below −1.5, so it is outside. Check: 2 × (−2)² − 5 × (−2) − 12 = 8 + 10 − 12 = 6, which is not below 0.

Rearrange It First

Get 0 on one side before you factorise.

Worked example

1
$6 x^{2} + 37 x \leq 35$ — not = 0 yet
2
$6 x^{2} + 37 x - 35 \leq 0$ — take 35 from both sides
3
$(6 x - 5) (x + 7) \leq 0$ — factorise
4
$- 7 \leq x \leq \frac{5}{6}$ — between the roots

Negative x²: Flip It First

Some questions put a minus in front of $x^{2}$ , like $12 + x - x^{2} > 0$ . The curve is upside down, so turn it the right way up first.

Worked example

1
$12 + x - x^{2} > 0$ — The question. The x² has a minus, so the curve is ∩-shaped.
2
$- x^{2} + x + 12 > 0$ — Write the terms in order: x² first, then x, then the number.
3
$x^{2} - x - 12 < 0$ — Multiply every term by −1. Multiplying by a negative flips the sign: > becomes <.
4
$(x - 4) (x + 3) < 0$ — Factorise. Check: −4x + 3x = −x, and (−4) × 3 = −12.
5
$x = 4 or x = - 3$ — The roots: each bracket equals 0.
6
$- 3 < x < 4$ — Now it is “< 0” for a U, so the answer is between the roots.

Key idea

Check with x = 0, which is inside the answer: 12 + 0 − 0 = 12, and 12 > 0. It works.

Common mistake

Forgetting to flip the sign. That gives x < −3 or x > 4, the exact opposite of the answer.

Won't Factorise? Use the Formula

Can't find the brackets? Use the quadratic formula to get the roots. The rest is the same: between or outside.

Worked example

1
$x^{2} - 4 x - 7 < 0$ — The question. No two whole numbers multiply to −7 and add to −4, so it will not factorise.
2
$a = 1, b = - 4, c = - 7$ — Read off a, b and c. Keep the minus signs.
3
$b^{2} - 4 a c = (- 4)^{2} - 4 \times 1 \times (- 7)$ — Work out the part under the square root. Put the negatives in brackets.
4
$= 16 + 28 = 44$ — (−4)² = 16, and −4 × 1 × (−7) = +28, because minus times minus is plus.
5
$x = \frac{- ( - 4 ) \pm 44}{2 \times 1}$ — Put it in the formula. −b means −(−4).
6
$x = \frac{4 \pm 44}{2}$ — −(−4) = 4, and 2 × 1 = 2.
7
$x = \frac{4 \pm 2 11}{2}$ — Simplify: √44 = √(4 × 11) = 2√11.
8
$x = 2 \pm 11$ — Divide both parts by 2. These are the two roots.
9
$2 - 11 < x < 2 + 11$ — Less than 0, so between the roots. Leave it exact unless the question asks for decimals.
10
$- 1.32 < x < 5.32$ — Only if decimals are asked: √11 = 3.3166…, so 2 − 3.3166 and 2 + 3.3166, to 3 s.f.

Key idea

Completing the square finds the same roots: $x^{2} - 4 x - 7 = (x - 2)^{2} - 11$ , so $x = 2 \pm 11$ .

From a Shape: Every Side Positive

The 5-mark version: a rectangle's area is less than a number. Every side must be positive, and that changes the answer.

Worked example

1
$(2 x + 3) (x - 1) < 75$ — Area = length × width, and it must be less than 75.
2
$2 x^{2} - 2 x + 3 x - 3 < 75$ — Expand: multiply each term in the first bracket by each term in the second.
3
$2 x^{2} + x - 3 < 75$ — Collect the x terms: −2x + 3x = x.
4
$2 x^{2} + x - 78 < 0$ — Take 75 from both sides: −3 − 75 = −78.
5
$(2 x + 13) (x - 6) < 0$ — Factorise. Check: −12x + 13x = x, and 13 × (−6) = −78.
6
$x = - 6.5 or x = 6$ — The roots: 2x + 13 = 0 gives −6.5, and x − 6 = 0 gives 6.
7
$- 6.5 < x < 6$ — Less than 0, so between the roots. This is not the final answer yet.
8
$x - 1 > 0 \Rightarrow x > 1$ — Check every side. A length cannot be negative or zero, so x − 1 > 0.
9
$2 x + 3 > 0 \Rightarrow x > - 1.5$ — The other side. Any x above 1 already makes this true, so it adds nothing.
10
$1 < x < 6$ — Keep only the part where every side is positive.

Common mistake

Stopping at −6.5 < x < 6. At x = 0 the side x − 1 would be −1, and a length cannot be negative.

Which Method? Read the Question

Roots first, then the sign.

You see	You do
< 0 or ≤ 0	Between the roots
> 0 or ≥ 0	Outside: two parts with “or”
Not = 0	Rearrange first
A minus in front of x²	Multiply by −1 and flip the sign
It won't factorise	Formula for the roots
List the integers	Whole numbers inside the answer
A shape with an area	Expand, solve, every side positive
A linear inequality	See Linear Inequalities

Key idea

Keep ≤ or ≥ if the question has them: the roots are included.

Key idea

Write the brackets and the roots down. The first marks are for finding the roots, so they still count if the final answer slips.

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