Quadratic Inequalities

What you'll learn

How to turn a quadratic inequality into an equation you can solve.
How roots split the number line into regions.
How to decide whether the answer is “between” the roots or “outside” the roots.
How to deal with rearranging, inclusive signs, coefficients, and integer solutions.

Quick recap: quadratics, factorising and roots

A quadratic expression is an expression where the highest power of $x$ is $x^2$ , such as $x^2-5x+6$ or $3x^2+7x-6$ .

To solve quadratic inequalities, you must first be confident solving the related quadratic equation.

Definition

Roots of a quadratic

The roots of a quadratic are the values of $x$ that make the quadratic equal to zero. On a graph, they are the points where the curve crosses or touches the x-axis.

A common way to find roots is by factorising. This means rewriting the quadratic as a product of two brackets.

Example

Finding the boundary values

Find the roots of $x^2-x-12=0$ .

Factorise the quadratic.
$x^2-x-12=(x-4)(x+3)$
Use the zero product rule: if two factors multiply to make zero, at least one factor must be zero.
$x−4=0⇒x=4x+3=0⇒x=−3\begin{aligned} x-4&=0 \Rightarrow x=4\\ x+3&=0 \Rightarrow x=-3 \end{aligned}$
The roots are $x = - 3$ and $x = 4$ .

What is a quadratic inequality?

Definition

Quadratic inequality

A quadratic inequality asks where a quadratic expression is greater than, less than, greater than or equal to, or less than or equal to zero. Examples include $x^2+4x+3>0$ and $2x^2-11x+9<0$ .

Instead of finding just the points where the quadratic equals zero, you are finding whole ranges of values of $x$ .

The roots are still important because they split the number line into regions. In each region, the quadratic is either positive or negative.

Example

Testing regions

Solve $(x - 2) (x - 6) < 0$ .

Find the boundary values by setting each bracket equal to zero.
$x−2=0⇒x=2x−6=0⇒x=6\begin{aligned} x-2&=0 \Rightarrow x=2\\ x-6&=0 \Rightarrow x=6 \end{aligned}$
The roots split the number line into three regions: $x < 2$ , $2 < x < 6$ , and $x > 6$ .
Test one value from each region.
- If $x = 0$ , then $(x - 2) (x - 6) = (- 2) (- 6) = 12$ , which is positive.
- If $x = 3$ , then $(x - 2) (x - 6) = (1) (- 3) = - 3$ , which is negative.
- If $x = 7$ , then $(x - 2) (x - 6) = (5) (1) = 5$ , which is positive.
The question asks for where the expression is less than zero, so choose the negative region.
$2 < x < 6$

The graph idea: inside or outside?

The graph of a quadratic is a U-shaped curve called a parabola. The coefficient of $x^2$ , often called the leading coefficient, is the number multiplying $x^2$ . If this coefficient is positive, the parabola opens upwards.

The diagram below shows the sign pattern for an upward-opening quadratic with two roots.

Upward-opening parabola showing positive outside the roots and negative between the roots

Key Idea

The key pattern

For an upward-opening quadratic with two roots, the quadratic is positive outside the roots and negative between the roots.

So for a positive $x^2$ coefficient:

$> 0$ or $≥0\ge 0$ usually means outside the roots.
$< 0$ or $≤0\le 0$ usually means between the roots.

Example

Choosing an outside region

Solve $x^2+7x+10>0$ .

Factorise the quadratic.
$x^2+7x+10=(x+5)(x+2)$
Find the roots.
$x+5=0⇒x=−5x+2=0⇒x=−2\begin{aligned} x+5&=0 \Rightarrow x=-5\\ x+2&=0 \Rightarrow x=-2 \end{aligned}$
The coefficient of $x^2$ is positive, so the parabola opens upwards.
The inequality asks for greater than zero, so choose the outside regions.
$\quad \text{or} \quad x>-2$

Common Mistake

Using and instead of or

For outside regions, you need two separate parts of the number line, so write “or”. For example, $x < - 5$ or $x > - 2$ , not one combined inequality.

Strict and inclusive inequalities

The inequality sign tells you whether to include the roots.

$<$ and $>$ are strict inequalities, so the roots are not included.
$≤\le$ and $≥\ge$ are inclusive inequalities, so the roots are included.

Think of the roots as the places where the expression equals zero. If the inequality allows equality, include them.

Example

Including the endpoints

Solve $x2−6x+8≤0x^2-6x+8\le 0$ .

Factorise the quadratic.
$x^2-6x+8=(x-2)(x-4)$
Find the roots.
$\quad \text{and} \quad x=4$
The parabola opens upwards, and the inequality asks for less than or equal to zero, so choose the region between the roots including the endpoints.
$2≤x≤42\le x\le 4$

Tip

Endpoint check

If the sign includes a line underneath, as in $≤\le$ or $≥\ge$ , your final answer should include the boundary values.

Rearranging first

Quadratic inequalities are easiest when one side is zero. This is called putting the inequality into standard form, such as $ax^2+bx+c<0$ .

You can add or subtract terms from both sides without changing the inequality sign. If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Example

Rearranging before solving

Solve $8x+20>x^2$ .

Move everything to one side, keeping the $x^2$ coefficient positive.
$0>x^2-8x-20$
Rewrite this with the quadratic on the left.
$x^2-8x-20<0$
Factorise.
$x^2-8x-20=(x-10)(x+2)$
The roots are $x = - 2$ and $x = 10$ .
The parabola opens upwards, and the inequality asks for less than zero, so choose the region between the roots.
$- 2 < x < 10$

Common Mistake

Forgetting to rearrange

Do not try to read the answer straight from something like $8x+20>x^2$ . First make one side zero, then solve the quadratic inequality.

When the coefficient of $x^2$ is not 1

Some quadratic inequalities have a coefficient in front of $x^2$ , such as $6x^2+11x-10<0$ . The method is the same, but factorising may be a little more careful.

Example

Factorising with a coefficient

Solve $4x^2-4x-15<0$ .

Factorise the quadratic.
$4x^2-4x-15=(2x-5)(2x+3)$
Find the roots.
$2x−5=0⇒x=522x+3=0⇒x=−32\begin{aligned} 2x-5&=0 \Rightarrow x=\frac{5}{2}\\ 2x+3&=0 \Rightarrow x=-\frac{3}{2} \end{aligned}$
Put the roots in order on the number line: $−32-\frac{3}{2}$ then $52\frac{5}{2}$ .
The coefficient of $x^2$ is positive, and the inequality asks for less than zero, so choose the region between the roots.
$−32<x<52-\frac{3}{2}<x<\frac{5}{2}$

Special case: inequalities with $x^2$

For inequalities like $x2≤64x^2\le 64$ , remember that both positive and negative values can square to make the same result. For example, both 8 and -8 square to 64.

Example

Solving an $x^{2}$ inequality

Solve $x2≤64x^2\le 64$ .

Move everything to one side.
$x2−64≤0x^2-64\le 0$
Factorise using the difference of two squares.
$x^2-64=(x-8)(x+8)$
The roots are $x = - 8$ and $x = 8$ .
The parabola opens upwards, and the inequality asks for less than or equal to zero, so choose the region between the roots including the endpoints.
$−8≤x≤8-8\le x\le 8$

Common Mistake

Only taking the positive square root

From $x2≤64x^2\le 64$ , do not write just $x≤8x\le 8$ . Negative values also work, so you need the full range $−8≤x≤8-8\le x\le 8$ .

Integer values and non-factorising quadratics

An integer is a whole number, such as -3, -2, -1, 0, 1, 2, 3.

Sometimes the quadratic does not factorise neatly. In that case, find the roots using the quadratic formula:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Then use the roots to decide which integers fit.

Example

Finding integer solutions

Work out the integer values satisfying $x^2-6x+7<0$ .

Find the roots by solving $x^2-6x+7=0$ . Here $a = 1$ , $b = - 6$ , and $c = 7$ .
$x=−(−6)±(−6)2−4(1)(7)2(1)x=\frac{-(-6)\pm\sqrt{(-6)^2-4(1)(7)}}{2(1)}$
Simplify the roots.
$x=6±36−282x=6±82x=3±2\begin{aligned} x&=\frac{6\pm\sqrt{36-28}}{2}\\ x&=\frac{6\pm\sqrt{8}}{2}\\ x&=3\pm\sqrt{2} \end{aligned}$
Estimate the roots: $3−2≈1.593-\sqrt{2}\approx 1.59$ and $3+2≈4.413+\sqrt{2}\approx 4.41$ .
The parabola opens upwards, and the inequality asks for less than zero, so the values must lie between the roots.
$3−2<x<3+23-\sqrt{2}<x<3+\sqrt{2}$
The integer values in this range are 2, 3 and 4.

Common Mistake

When there are no roots

If a quadratic has no real roots, the graph never crosses the x-axis. Test one value of $x$ , such as $x = 0$ , to decide whether the inequality is true for all values of $x$ or for no values.

Exam technique

In the exam

Rearrange first so one side is zero, preferably with a positive $x^2$ coefficient.
Find the roots carefully, then mark them in order on a quick number line.
Decide between “inside” and “outside”, and check whether the inequality sign includes the endpoints.

Self review

Check yourself

Can you explain why $x^2-9>0$ gives two separate regions?
When should the roots be included in your final answer?
If a quadratic inequality asks for integer values, why might you need to estimate the roots first?

Quadratic Inequalities

What you'll learn

How to turn a quadratic inequality into an equation you can solve.
How roots split the number line into regions.
How to decide whether the answer is “between” the roots or “outside” the roots.
How to deal with rearranging, inclusive signs, coefficients, and integer solutions.

Quick recap: quadratics, factorising and roots

A quadratic expression is an expression where the highest power of $x$ is $x^2$ , such as $x^2-5x+6$ or $3x^2+7x-6$ .

To solve quadratic inequalities, you must first be confident solving the related quadratic equation.

Definition

Roots of a quadratic

The roots of a quadratic are the values of $x$ that make the quadratic equal to zero. On a graph, they are the points where the curve crosses or touches the x-axis.

A common way to find roots is by factorising. This means rewriting the quadratic as a product of two brackets.

Example

Finding the boundary values

Find the roots of $x^2-x-12=0$ .

Factorise the quadratic.
$x^2-x-12=(x-4)(x+3)$
Use the zero product rule: if two factors multiply to make zero, at least one factor must be zero.
$x−4=0⇒x=4x+3=0⇒x=−3\begin{aligned} x-4&=0 \Rightarrow x=4\\ x+3&=0 \Rightarrow x=-3 \end{aligned}$
The roots are $x = - 3$ and $x = 4$ .

What is a quadratic inequality?

Definition

Quadratic inequality

A quadratic inequality asks where a quadratic expression is greater than, less than, greater than or equal to, or less than or equal to zero. Examples include $x^2+4x+3>0$ and $2x^2-11x+9<0$ .

Instead of finding just the points where the quadratic equals zero, you are finding whole ranges of values of $x$ .

The roots are still important because they split the number line into regions. In each region, the quadratic is either positive or negative.

Example

Testing regions

Solve $(x - 2) (x - 6) < 0$ .

Find the boundary values by setting each bracket equal to zero.
$x−2=0⇒x=2x−6=0⇒x=6\begin{aligned} x-2&=0 \Rightarrow x=2\\ x-6&=0 \Rightarrow x=6 \end{aligned}$
The roots split the number line into three regions: $x < 2$ , $2 < x < 6$ , and $x > 6$ .
Test one value from each region.
- If $x = 0$ , then $(x - 2) (x - 6) = (- 2) (- 6) = 12$ , which is positive.
- If $x = 3$ , then $(x - 2) (x - 6) = (1) (- 3) = - 3$ , which is negative.
- If $x = 7$ , then $(x - 2) (x - 6) = (5) (1) = 5$ , which is positive.
The question asks for where the expression is less than zero, so choose the negative region.
$2 < x < 6$

The graph idea: inside or outside?

The diagram below shows the sign pattern for an upward-opening quadratic with two roots.

Upward-opening parabola showing positive outside the roots and negative between the roots

Key Idea

The key pattern

For an upward-opening quadratic with two roots, the quadratic is positive outside the roots and negative between the roots.

So for a positive $x^2$ coefficient:

$> 0$ or $≥0\ge 0$ usually means outside the roots.
$< 0$ or $≤0\le 0$ usually means between the roots.

Example

Choosing an outside region

Solve $x^2+7x+10>0$ .

Factorise the quadratic.
$x^2+7x+10=(x+5)(x+2)$
Find the roots.
$x+5=0⇒x=−5x+2=0⇒x=−2\begin{aligned} x+5&=0 \Rightarrow x=-5\\ x+2&=0 \Rightarrow x=-2 \end{aligned}$
The coefficient of $x^2$ is positive, so the parabola opens upwards.
The inequality asks for greater than zero, so choose the outside regions.
$\quad \text{or} \quad x>-2$

Common Mistake

Using and instead of or

For outside regions, you need two separate parts of the number line, so write “or”. For example, $x < - 5$ or $x > - 2$ , not one combined inequality.

Strict and inclusive inequalities

The inequality sign tells you whether to include the roots.

$<$ and $>$ are strict inequalities, so the roots are not included.
$≤\le$ and $≥\ge$ are inclusive inequalities, so the roots are included.

Think of the roots as the places where the expression equals zero. If the inequality allows equality, include them.

Example

Including the endpoints

Solve $x2−6x+8≤0x^2-6x+8\le 0$ .

Factorise the quadratic.
$x^2-6x+8=(x-2)(x-4)$
Find the roots.
$\quad \text{and} \quad x=4$
The parabola opens upwards, and the inequality asks for less than or equal to zero, so choose the region between the roots including the endpoints.
$2≤x≤42\le x\le 4$

Tip

Endpoint check

If the sign includes a line underneath, as in $≤\le$ or $≥\ge$ , your final answer should include the boundary values.

Rearranging first

Quadratic inequalities are easiest when one side is zero. This is called putting the inequality into standard form, such as $ax^2+bx+c<0$ .

You can add or subtract terms from both sides without changing the inequality sign. If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Example

Rearranging before solving

Solve $8x+20>x^2$ .

Move everything to one side, keeping the $x^2$ coefficient positive.
$0>x^2-8x-20$
Rewrite this with the quadratic on the left.
$x^2-8x-20<0$
Factorise.
$x^2-8x-20=(x-10)(x+2)$
The roots are $x = - 2$ and $x = 10$ .
The parabola opens upwards, and the inequality asks for less than zero, so choose the region between the roots.
$- 2 < x < 10$

Common Mistake

Forgetting to rearrange

Do not try to read the answer straight from something like $8x+20>x^2$ . First make one side zero, then solve the quadratic inequality.

When the coefficient of $x^2$ is not 1

Some quadratic inequalities have a coefficient in front of $x^2$ , such as $6x^2+11x-10<0$ . The method is the same, but factorising may be a little more careful.

Example

Factorising with a coefficient

Solve $4x^2-4x-15<0$ .

Factorise the quadratic.
$4x^2-4x-15=(2x-5)(2x+3)$
Find the roots.
$2x−5=0⇒x=522x+3=0⇒x=−32\begin{aligned} 2x-5&=0 \Rightarrow x=\frac{5}{2}\\ 2x+3&=0 \Rightarrow x=-\frac{3}{2} \end{aligned}$
Put the roots in order on the number line: $−32-\frac{3}{2}$ then $52\frac{5}{2}$ .
The coefficient of $x^2$ is positive, and the inequality asks for less than zero, so choose the region between the roots.
$−32<x<52-\frac{3}{2}<x<\frac{5}{2}$

Special case: inequalities with $x^2$

For inequalities like $x2≤64x^2\le 64$ , remember that both positive and negative values can square to make the same result. For example, both 8 and -8 square to 64.

Example

Solving an $x^{2}$ inequality

Solve $x2≤64x^2\le 64$ .

Move everything to one side.
$x2−64≤0x^2-64\le 0$
Factorise using the difference of two squares.
$x^2-64=(x-8)(x+8)$
The roots are $x = - 8$ and $x = 8$ .
The parabola opens upwards, and the inequality asks for less than or equal to zero, so choose the region between the roots including the endpoints.
$−8≤x≤8-8\le x\le 8$

Common Mistake

Only taking the positive square root

From $x2≤64x^2\le 64$ , do not write just $x≤8x\le 8$ . Negative values also work, so you need the full range $−8≤x≤8-8\le x\le 8$ .

Integer values and non-factorising quadratics

An integer is a whole number, such as -3, -2, -1, 0, 1, 2, 3.

Sometimes the quadratic does not factorise neatly. In that case, find the roots using the quadratic formula:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Then use the roots to decide which integers fit.

Example

Finding integer solutions

Work out the integer values satisfying $x^2-6x+7<0$ .

Find the roots by solving $x^2-6x+7=0$ . Here $a = 1$ , $b = - 6$ , and $c = 7$ .
$x=−(−6)±(−6)2−4(1)(7)2(1)x=\frac{-(-6)\pm\sqrt{(-6)^2-4(1)(7)}}{2(1)}$
Simplify the roots.
$x=6±36−282x=6±82x=3±2\begin{aligned} x&=\frac{6\pm\sqrt{36-28}}{2}\\ x&=\frac{6\pm\sqrt{8}}{2}\\ x&=3\pm\sqrt{2} \end{aligned}$
Estimate the roots: $3−2≈1.593-\sqrt{2}\approx 1.59$ and $3+2≈4.413+\sqrt{2}\approx 4.41$ .
The parabola opens upwards, and the inequality asks for less than zero, so the values must lie between the roots.
$3−2<x<3+23-\sqrt{2}<x<3+\sqrt{2}$
The integer values in this range are 2, 3 and 4.

Common Mistake

When there are no roots

If a quadratic has no real roots, the graph never crosses the x-axis. Test one value of $x$ , such as $x = 0$ , to decide whether the inequality is true for all values of $x$ or for no values.

Exam technique

In the exam

Rearrange first so one side is zero, preferably with a positive $x^2$ coefficient.
Find the roots carefully, then mark them in order on a quick number line.
Decide between “inside” and “outside”, and check whether the inequality sign includes the endpoints.

Self review

Check yourself

Can you explain why $x^2-9>0$ gives two separate regions?
When should the roots be included in your final answer?
If a quadratic inequality asks for integer values, why might you need to estimate the roots first?

Quadratic Inequalities

What you'll learn

Quick recap: quadratics, factorising and roots

What is a quadratic inequality?

The graph idea: inside or outside?

Strict and inclusive inequalities

Rearranging first

When the coefficient of $x^2$ is not 1

Special case: inequalities with $x^2$

Integer values and non-factorising quadratics

Recap questions

You've reached the end

Quadratic Inequalities

What you'll learn

Quick recap: quadratics, factorising and roots

What is a quadratic inequality?

The graph idea: inside or outside?

Strict and inclusive inequalities

Rearranging first

When the coefficient of $x^2$ is not 1

Special case: inequalities with $x^2$

Integer values and non-factorising quadratics

Recap questions

You've reached the end

Quadratic Inequalities

What you'll learn

Quick recap: quadratics, factorising and roots

What is a quadratic inequality?

The graph idea: inside or outside?

Strict and inclusive inequalities

Rearranging first

When the coefficient of x2x^2x2 is not 1

Special case: inequalities with x2x^2x2

Integer values and non-factorising quadratics

Recap questions

You've reached the end

Quadratic Inequalities Revision Guide

Quadratic Inequalities

What you'll learn

Quick recap: quadratics, factorising and roots

What is a quadratic inequality?

The graph idea: inside or outside?

Strict and inclusive inequalities

Rearranging first

When the coefficient of x2x^2x2 is not 1

Special case: inequalities with x2x^2x2

Integer values and non-factorising quadratics

Recap questions

You've reached the end

Quadratic Inequalities Revision Guide

When the coefficient of $x^2$ is not 1

Special case: inequalities with $x^2$

When the coefficient of $x^2$ is not 1

Special case: inequalities with $x^2$