Question 1 Easy2 marks
Solve x2-4>0.
Show answer
Answer: x<-2 or x>2
- Same boundaries -2 and 2.
- The square exceeds 4 outside them.
- Method (1): Same boundaries -2 and 2
- Accuracy (1): Answer x<-2 or x>2
Quadratic inequalities practice questions by difficulty
Here you practise quadratic inequalities; the teaching is elsewhere. Below are 4 questions (2 warm-up, 2 core, 0 stretch, 1 to 2 marks each), all new to you if you read the topic page. Also here: a multi-part show-that question, 15 quick checks. If a question beats you, go back to the Quadratic inequalities topic page or the Quadratic inequalities worked examples.
How to use this page: Try the stretch set first; if you drop marks, come back down to core and find which step went wrong.
Warm-up questions
2 questions, 1 to 2 marks each. Leading off: “Solve x2-4>0.”
Question 2 Easy1 mark
Solve (x+1)(x-3)<0.
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Answer: -1<x<3
- Roots -1 and 3.
- Negative between them.
- Answer (1): Answer -1<x<3
Core questions
2 questions, 1 to 2 marks each. Opening with: “Solve x2+2x-3>0.”
Question 3 Medium2 marks
Solve x2+2x-3>0.
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Answer: x<-3 or x>1
- (x+3)(x-1)=0.
- Positive outside -3 and 1.
- Method (1): (x+3)(x-1)=0
- Accuracy (1): Answer x<-3 or x>1
Question 4 Medium1 mark
Solve x2+2x-3<=0.
Show answer
Answer: -3<=x<=1
- Same roots -3 and 1.
- Inclusive between them.
- Answer (1): Answer -3<=x<=1
Multi-part exam-style question
Exam style: 2 linked parts worth 3 marks in total. Set by Yumio in the GCSE question style.
Question 5 Exam-style · Show that3 marks
Part (a) is a show that question: the answer is printed and the method earns the marks. Show your working throughout.
(a) The inequality x2>9 has which solution? Show that the answer is x<-3 or x>3. [2 marks]
(b) Solve (x-1)(x-4)<=0 and say whether 1 is included. [1 mark]
Show answer and worked solution
Part (a)
Answer: Shown: the working reaches x<-3 or x>3
- Roots -3 and 3.
- A square bigger than 9 sits outside them.
Marks for this part:
- M1: a correct first step towards x<-3 or x>3
- A1: reaches x<-3 or x>3 with every line correct
Part (b)
Answer: 1<=x<=4
- The product is negative between 1 and 4, and zero at the ends.
- 1 satisfies the equals case.
Marks for this part:
- B1: Answer 1<=x<=4
Quick checks
Short one-step questions. Pick an option before you open the answer.
Check 1
Where is a positive parabola negative?
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Between its two roots. Between its two roots.
Check 2
Where is a positive parabola positive?
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Outside its two roots. Outside its two roots.
Check 3
What is the difference between x2-5x+6<0 and x2-5x+6=0?
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The equation gives the boundaries. The inequality gives a region. The equation gives the boundaries. The inequality gives a region.
Check 4
Does a strict inequality include the roots?
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No. The roots make the expression zero, not negative or positive. No. The roots make the expression zero, not negative or positive.
Check 5
When does the solution include the roots?
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When the sign is less than or equal, or greater than or equal. When the sign is less than or equal, or greater than or equal.
Check 6
Why can you divide 2x2-8x+6<0 by 2?
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2 is positive, so the inequality sign stays the same. 2 is positive, so the inequality sign stays the same.
Check 7
What would a negative divisor do to the sign?
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It would flip the inequality. Dividing by a negative is a different step. It would flip the inequality. Dividing by a negative is a different step.
Check 8
How do you write an outside region?
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Two inequalities joined by or. Two inequalities joined by or.
Check 9
How do you write an inside region?
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One double inequality, smaller root, then x, then larger root. One double inequality, smaller root, then x, then larger root.
Check 10
The roots are -2 and 2 and the curve is above the axis outside. Which inequality is that for y=x2-4?
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x<-2 or x>2. x<-2 or x>2.
Check 11
Why sketch the U-shape?
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It shows inside versus outside without testing lots of numbers. It shows inside versus outside without testing lots of numbers.
Check 12
A test point x=0 in x2-5x+6 gives +6. What does that say?
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Zero is not in the less-than region. The region is between 2 and 3. Zero is not in the less-than region. The region is between 2 and 3.
Check 13
Can a quadratic inequality have one root only as its solution?
Show answer
Only in special cases, such as a square that is never negative. The usual two-root case has an interval or two rays. Only in special cases, such as a square that is never negative. The usual two-root case has an interval or two rays.
Check 14
(x-2)(x-3)<0 is true for which kind of x?
Show answer
x between 2 and 3. x between 2 and 3.
Check 15
Name the solution of x2-4<0.
Show answer
-2<x<2. The square is less than 4 between -2 and 2. -2<x<2. The square is less than 4 between -2 and 2.
Keep going with Quadratic inequalities
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