Common Mistakes with Quadratic Inequalities in GCSE Maths

A quadratic inequality can look almost identical to a quadratic equation. That is why it catches students out. You factorise, find two correct roots and feel that the difficult work is finished. Then the final inequality is written the wrong way round, costing the marks that mattered.

Understanding the common mistakes with quadratic inequalities in GCSE Maths means learning what happens after you find the roots. You must decide where the quadratic is positive or negative, include boundaries when equality is allowed, and express the answer as an interval rather than two isolated values.

Imagine a student checking a marked practice paper. Their algebra is often sound. The errors happen in the final few lines. We will follow that student as they identify each mistake, understand why it happened and replace it with a dependable method.

The quadratic inequalities checklist

For an inequality such as $x^2-x-6<0$, use this process:

• Rearrange so that one side is $0$.
• Solve the related equation to find the roots.
• Sketch or visualise the quadratic graph.
• Identify where the graph satisfies the inequality.
• Check whether the boundary values are included.
• Write the complete solution using inequalities and, where needed, the word “or”.

The key idea is graphical. If $f(x)>0$, you want the parts of the graph above the $x$-axis. If $f(x)<0$, you want the parts below it.

The matching Quadratic Inequalities lesson and resources provide explanations and questions for practising this full process.

Mistake: stopping after finding the roots

Consider:

$$x^2-x-6<0$$

A student might write:

$$x=-2orx=3$$

Those are the roots of the related equation, but they are not the solution of the inequality. The roots are only the boundaries of the regions we need to investigate.

Correct method

First factorise:

$$x^2-x-6=(x-3)(x+2)$$

Set the quadratic equal to $0$ to find its roots:

$$(x-3)(x+2)=0$$ $$x=3orx=-2$$

Because the coefficient of $x^2$ is positive, the graph is a U-shaped parabola. It lies below the $x$-axis between its roots. Therefore:

$$-2<x<3$$

The student’s roots were correct, but the question was asking for every value of $x$ that makes the expression negative.

If factorising is the uncertain step, revisit the Solving Quadratics exam booklet before combining the technique with inequalities.

Mistake: choosing inside or outside without checking

Some students memorise that the answer is “between the roots”. That works for an upward-opening quadratic when the expression must be negative, but it is not a universal rule.

Solve:

$$x^2+x-6\ge 0$$

Factorising gives:

$$(x+3)(x-2)\ge 0$$

The roots are $x=-3$ and $x=2$. The graph opens upwards, but this time we want the parts on or above the $x$-axis. These occur outside the roots:

$$x\le -3orx\ge 2$$

A quick sketch prevents guesswork. Mark the roots on the $x$-axis, draw the general shape of the parabola and shade the required region mentally or on paper.

You now have the complete method. If setting up repeated practice feels like another task, use the Maths Genie GCSE Maths Lessons and Question Bank to move directly from an explanation to focused quadratic inequalities questions.

Mistake: ignoring a negative leading coefficient

Now consider:

$$-x^2+5x-6>0$$

The negative coefficient of $x^2$ means the graph opens downwards. A student who automatically draws a U-shape may select the wrong region.

One approach is to multiply the whole inequality by $-1$. When an inequality is multiplied or divided by a negative number, its sign reverses:

$$x^2-5x+6<0$$

Factorise:

$$(x-2)(x-3)<0$$

The upward-opening graph is below the axis between the roots, so:

$$2<x<3$$

Alternatively, keep the original expression. Its downward-opening graph is above the axis between $x=2$ and $x=3$, giving the same answer.

The sign reversal is not a minor notation detail. It changes the question. Writing the intermediate line makes the change visible and easier to check.

Mistake: including or excluding the boundary incorrectly

The symbols $<$ and $>$ exclude the roots because the expression must be strictly negative or positive. The symbols $\le$ and $\ge$ include roots because equality is allowed.

For example:

$$(x-1)(x-4)\le 0$$

The graph is below the axis between the roots and equals $0$ at the roots. Therefore:

$$1\le x\le 4$$

If the original inequality had been

$$(x-1)(x-4)<0,$$

then the solution would be:

$$1<x<4$$

A useful final check is to substitute a boundary into the original expression. At $x=1$, the expression equals $0$. Ask whether $0$ is accepted by the inequality symbol.

Mistake: testing only one value

A quadratic with two different roots divides the number line into three regions. Testing one value does not tell you what happens in all three.

For $(x+1)(x-5)>0$, the roots are $-1$ and $5$. The regions are:

• $x<-1$
• $-1<x<5$
• $x>5$

Test one convenient value from each region. Using $x=-2$, $x=0$ and $x=6$ shows that the expression is positive in the two outside regions. Hence:

$$x<-1orx>5$$

A sketch is normally quicker, but region testing is a reliable checking method. It is especially useful when a factorised expression or negative leading coefficient makes the graph less immediately obvious.

For more support with the underlying equation skills, the Quadratic Equations revision guide reviews factorising and the main solving methods.

Mistake: giving disconnected regions as one interval

Suppose the correct regions are $x<-4$ and $x>1$. Writing

$$-4<x<1$$

describes the exact opposite region. It joins the boundaries instead of recording the two separate outer sections.

The correct answer is:

$$x<-4orx>1$$

Use “and” when one interval lies between two boundaries, as in $-4<x<1$. Use “or” when either of two separate regions works.

Exam-style practice

Try these without looking back at the worked examples. Factorise, sketch the curve and state the full solution.

Question A

Solve:

$$x^2-7x+10<0$$

The factorisation is $(x-2)(x-5)$. The graph opens upwards and is negative between its roots:

$$2<x<5$$

Question B

Solve:

$$x^2-2x-8\ge 0$$

The factorisation is $(x-4)(x+2)$. The graph is on or above the axis outside the roots:

$$x\le -2orx\ge 4$$

Question C

Solve:

$$-x^2+4x+5\le 0$$

Multiplying by $-1$ and reversing the sign gives:

$$x^2-4x-5\ge 0$$

Factorising gives $(x-5)(x+1)\ge 0$, so:

$$x\le -1orx\ge 5$$

After each question, check four things: the roots, the curve’s direction, the required region and the boundary symbols. That brief routine catches most quadratic inequalities mistakes before they cost marks.

Turn correct methods into reliable exam marks

Knowing the method once is different from recognising it under exam pressure. Your next step is to practise choosing the correct regions until the sketch, boundaries and final inequality become one connected process. Start the Maths Genie Quadratic Inequalities Lesson, then answer focused questions in the GCSE Maths Question Bank.