Mathematics · Year 10 · Algebraic Structure and Manipulation · Autumn Term

Quadratic Inequalities

Solving quadratic inequalities and representing solution sets on number lines and graphs.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

About This Topic

Quadratic inequalities ask students to solve expressions such as ax² + bx + c > 0 by finding roots of the related equation, then identifying intervals where the quadratic is positive or negative. Year 10 students sketch parabolas to see where they lie above or below the x-axis, represent solutions on number lines with open or closed circles, and handle cases with no real roots or repeated roots. This extends quadratic equation skills into inequalities, using factorisation, completing the square, or the discriminant.

In the GCSE algebra curriculum, this topic connects algebraic manipulation to graphical interpretation and logical reasoning. Students explain why solution sets form one or two intervals, unlike linear inequalities' single split, and construct inequalities matching specific intervals. These steps build precision in notation and confidence in multi-step problems, preparing for higher maths like calculus inequalities.

Active learning suits quadratic inequalities well. When students in small groups test points on interactive graphs or build physical sign charts with string and markers, they visualise parabola shapes and sign changes. Collaborative verification catches errors quickly, turning abstract rules into shared discoveries that stick.

Key Questions

1. Explain the graphical approach to solving quadratic inequalities.
2. Differentiate between the solution sets of linear and quadratic inequalities.
3. Construct a quadratic inequality whose solution is a single interval.

Learning Objectives

• Analyze the graphical representation of quadratic inequalities to determine solution intervals.
• Compare the solution sets of quadratic inequalities with those of linear inequalities, identifying key differences in interval notation.
• Construct a quadratic inequality given a specific solution set represented on a number line.
• Calculate the roots of the related quadratic equation to define the boundaries of the solution set for a quadratic inequality.
• Explain the relationship between the sign of the quadratic expression and the intervals above or below the x-axis.

Before You Start

Solving Quadratic Equations

Why: Students must be able to find the roots of a quadratic equation using factorization, completing the square, or the quadratic formula to identify the boundary points for inequality solutions.

Graphing Linear and Quadratic Functions

Why: Understanding the shape and behavior of parabolas is essential for interpreting the graphical method of solving quadratic inequalities.

Solving Linear Inequalities

Why: Familiarity with inequality symbols and representing solution sets on a number line provides a foundation for more complex quadratic inequalities.

Key Vocabulary

Quadratic Inequality	An inequality involving a quadratic expression, such as ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0.
Parabola	The U-shaped graph of a quadratic function, which opens upwards or downwards depending on the sign of the leading coefficient.
Roots (of a quadratic equation)	The values of x for which the related quadratic equation ax² + bx + c = 0 equals zero; these are the points where the parabola intersects the x-axis.
Solution Interval	A continuous range of values on the number line that satisfies the given inequality.

Watch Out for These Misconceptions

Common Misconception

The solution is always between the roots.

What to Teach Instead

Solutions depend on the parabola's direction: for a > 0, outside roots if positive leading coefficient; inside if negative. Group testing of points in intervals reveals this pattern quickly. Peer explanations during matching activities solidify the rule.

Common Misconception

Treat quadratic inequalities exactly like equations by equating to zero.

What to Teach Instead

Equations give points, but inequalities need ranges determined by signs. Hands-on sign charts in pairs help students plot roots and test, distinguishing boundaries from regions. Discussion exposes why equality signs matter for closed intervals.

Common Misconception

Number line solutions are always a single interval like linear ones.

What to Teach Instead

Quadratics can yield two intervals or none due to the curve. Collaborative graph shading shows disconnected regions clearly. Students revise personal models through group galleries of examples.

Active Learning Ideas

Pairs: Graph and Number Line Match-Up

Provide cards with quadratic inequalities, graphs, and number lines. Pairs match sets correctly, test points to verify, and justify choices. Pairs swap sets with neighbours for peer review.

30 min·Pairs

Small Groups: Sign Chart Construction

Groups receive quadratics to factorise, draw sign charts on large paper, and shade solution regions. Each member tests an interval point. Groups present one to the class for critique.

40 min·Small Groups

Whole Class: Interval Challenge Relay

Display target intervals on the board. Teams send one student at a time to construct and solve a quadratic inequality matching it, passing a marker. First accurate team wins.

25 min·Whole Class

Individual: Digital Graph Explorer

Students use graphing software to input quadratics, adjust sliders for a and b, observe solution changes, and screenshot three examples with explanations for a class gallery.

35 min·Individual

Real-World Connections

• Engineers designing suspension bridges use quadratic equations to model the shape of the main cables. Quadratic inequalities can then be used to define safe load limits or stress tolerances within specific sections of the bridge structure.
• Financial analysts model investment growth using quadratic functions. Quadratic inequalities help determine the time periods or investment amounts for which the investment value will exceed a certain target or remain below a risk threshold.

Assessment Ideas

Exit Ticket

Provide students with the inequality x² - 5x + 6 > 0. Ask them to: 1. Find the roots of x² - 5x + 6 = 0. 2. Sketch a graph of y = x² - 5x + 6. 3. Shade the solution interval on a number line.

Quick Check

Display two inequalities on the board: 1) 2x + 3 < 7 and 2) x² + x - 2 > 0. Ask students to write down the type of inequality, the general shape of its graph (if applicable), and whether its solution set is typically a single interval or two intervals.

Discussion Prompt

Pose the question: 'How does the graphical approach help us understand why the solution to a quadratic inequality like x² - 4 < 0 is a single interval, while the solution to x² - 4 > 0 is two separate intervals?' Encourage students to refer to sketches of the parabola y = x² - 4.

Frequently Asked Questions

How do you solve quadratic inequalities graphically?

Sketch the parabola y = ax² + bx + c, mark roots, and shade where it meets the inequality: above x-axis for > 0 if a > 0. Test points confirm intervals. Number lines show solutions with circles: open for strict, closed for inclusive. Practice with varied a values builds fluency across 10-15 examples.

What are common mistakes with quadratic inequalities?

Students often ignore the leading coefficient's effect on sign regions or forget to flip for < 0 cases. They may treat roots as full solutions or misuse discriminant for complex roots. Address with paired checklists: factorise, chart signs, test points. Regular low-stakes quizzes track progress.

How do quadratic inequalities differ from linear ones?

Linear inequalities split the line into one interval; quadratics, due to parabolas, create zero, one, or two intervals. Graphs show lines crossing once versus curves potentially twice. Students construct examples: linear x > 3 is simple, but x² - 4 > 0 is x < -2 or x > 2, highlighting non-linear behaviour.

What active learning strategies work for quadratic inequalities?

Use pair match-ups of inequalities to graphs and number lines for quick feedback. Small group sign chart builds with string models visualise changes. Whole-class relays for constructing inequalities from intervals promote competition and precision. These methods make abstract signs tangible, reduce errors through talk, and fit 30-40 minute slots effectively.