Quadratic Inequalities – Grade 10 IGCSE Maths

Welcome to Quadratic Inequalities!

A quadratic inequality asks: for which values of x is a parabola above (or below) the x-axis? The key skill is sketching the parabola, finding the critical values (roots), and then deciding whether the solution is BETWEEN or OUTSIDE the roots.

a>0, ax²+bx+c<0 → BETWEEN roots: α < x < β | a>0, >0 → OUTSIDE: x < α or x > β

The 4-Step Method

Rearrange, factor, sketch, decide

Set Notation & Integers

∪, ∩ and integer solutions

Negative a & Special Cases

Sign flip, always true/false, combined

Worked Examples

6 step-by-step examples

Visualiser

Sliders plot and shade parabola

Exercises

5 sets of 8 questions + Practice + Challenge

1. Solving Quadratic Inequalities — The 4-Step Method

Step 1: Rearrange → Step 2: Find roots → Step 3: Sketch parabola → Step 4: Apply the KEY RULE

Solve x² − 5x + 6 > 0:
Step 1: Already in ax²+bx+c>0 form ✓
Step 2: x²−5x+6 = (x−2)(x−3) → roots: x=2 and x=3
Step 3: a=1>0 → U-shaped parabola, crossing x-axis at 2 and 3
Step 4: We want >0 (above x-axis). For U-shape, this is OUTSIDE the roots.
Answer: x < 2 or x > 3

Solve x² − 5x + 6 < 0:
Same roots: 2 and 3. Now want <0 (below x-axis). For U-shape, BETWEEN roots.
Answer: 2 < x < 3

THE KEY RULE (a > 0 always):
Quadratic < 0 → BETWEEN roots: α < x < β (the "valley")
Quadratic > 0 → OUTSIDE roots: x < α or x > β (the "hills")

Memory: "Less than → between; Greater than → outer"
Think of the U-shape: the trough (below zero) is BETWEEN the roots. The arms (above zero) are OUTSIDE the roots.

More Examples of Step 1 — Rearranging First

Solve x² − 3x > 10:
Rearrange: x² − 3x − 10 > 0
Factor: (x−5)(x+2) > 0 → roots: 5 and −2
a>0, >0 → OUTSIDE: x < −2 or x > 5

Solve 2x² − 8 < 0:
2x² < 8 → x² < 4 → (x−2)(x+2) < 0
Roots: −2 and 2. a>0, <0 → BETWEEN: −2 < x < 2

2. Set Notation, Number Lines and Integer Solutions

Set notation:
"x < 2 or x > 3" → {x : x<2} ∪ {x : x>3}
"2 < x < 3" → {x : 2<x<3}
The symbol ∪ means "or" (union). Use ∩ for "and" (intersection).

Integer solutions:
Solve x²−5x+4 < 0 → (x−1)(x−4) < 0 → 1 < x < 4
Integers in this range (NOT including endpoints): {2, 3}
Number of integer solutions = 2

Double inequality — solving n² ≤ 25:
n² ≤ 25 → (n−5)(n+5) ≤ 0 → roots: −5 and 5
a>0, ≤0 → BETWEEN (including endpoints): −5 ≤ n ≤ 5

Number line: Use open circles ○ for strict inequalities (< or >) and closed circles ● for non-strict (≤ or ≥). Shade the solution region.

3. Negative Coefficient and Special Cases

Case: a < 0 (inverted parabola ∩)
Multiply both sides by −1 and FLIP the inequality sign.
Solve −x² + 4x − 3 > 0
Multiply by −1: x² − 4x + 3 < 0 (sign flipped)
Factor: (x−1)(x−3) < 0 → BETWEEN: 1 < x < 3

Remember: Multiplying or dividing an inequality by a NEGATIVE number REVERSES the inequality sign. This is the most common error with negative-a quadratics.

No real roots (discriminant < 0):
x² + x + 3 > 0: discriminant = 1−12 = −11 < 0, no real roots.
Parabola never crosses x-axis. Since a>0, it is ALWAYS above x-axis.
Solution: all real x (x ∈ ℝ)

Conversely, x² + x + 3 < 0 has NO solution (parabola never goes below x-axis).

Combined inequalities:
Solve: x² − 9 > 0 AND x < 10
x² − 9 > 0 → (x−3)(x+3) > 0 → x < −3 or x > 3
Intersect with x < 10: x < −3 or 3 < x < 10

Strategy for combined: Solve each inequality separately, then find the intersection (overlap) of the solution sets. Drawing number lines for each helps enormously.

Example 1 — Standard > 0

Solve x² − 7x + 10 > 0

Factor: (x−2)(x−5) > 0. Roots: 2 and 5.

a>0, >0 → OUTSIDE: x < 2 or x > 5

Example 2 — Negative a

Solve −x² + 6x − 5 < 0

Multiply by −1 (flip sign): x² − 6x + 5 > 0

Factor: (x−1)(x−5) > 0 → OUTSIDE: x < 1 or x > 5

Example 3 — Integer Solutions

Solve x² − 6x + 8 < 0 and list integer solutions

Factor: (x−2)(x−4) < 0 → BETWEEN: 2 < x < 4

Integer solutions: {3} — only x=3 is strictly between 2 and 4.

Example 4 — Always True

Solve x² + 4x + 5 > 0

Discriminant: 16 − 20 = −4 < 0. No real roots. a>0.

Parabola is always above x-axis. Solution: all real x

Example 5 — Combined with Linear

Solve x² − 4 > 0 AND x < 5

(x−2)(x+2)>0 → x<−2 or x>2. Intersect with x<5.

x < −2 or 2 < x < 5

Example 6 — Rearrange First

Solve x² > 3x + 4

Rearrange: x² − 3x − 4 > 0

Factor: (x−4)(x+1) > 0 → OUTSIDE: x < −1 or x > 4

Common Mistakes

Mistake 1: Outside vs Between confusion
For ax²+bx+c < 0 (a>0), the answer is BETWEEN the roots. For > 0, it's OUTSIDE. Many students do this backwards. Sketch the parabola every time to see which region is above/below zero.
Mistake 2: Forgetting to flip the sign for negative a
When you multiply −x²+... by −1, the inequality sign MUST flip. −x²>0 becomes x²<0, not x²>0.
Mistake 3: Including endpoints for strict inequalities
x²−4 > 0 → x < −2 or x > 2 (strict, open circles). x²−4 ≥ 0 → x ≤ −2 or x ≥ 2 (non-strict, closed circles). Know which to use.
Mistake 4: Integer solutions at endpoints
For 1 < x < 4, x=1 and x=4 are NOT integer solutions. Only integers strictly between the roots count.
Mistake 5: Combined intervals — wrong intersection
When combining two inequalities with AND, you need the INTERSECTION (overlap), not the union. Draw both solutions on separate number lines and shade the overlap.

Key Formulas & Rules Summary

Form	Rule	Solution
ax²+bx+c < 0 (a>0)	Below parabola, between roots	α < x < β
ax²+bx+c > 0 (a>0)	Above parabola, outside roots	x < α or x > β
ax²+bx+c < 0 (a<0)	Multiply by −1, flip sign, re-apply rule	x < α or x > β
Discriminant < 0, a>0, >0	Parabola entirely above x-axis	All real x
Discriminant < 0, a>0, <0	Parabola entirely above x-axis	No solution
Set notation "or"	x < α or x > β	{x:x<α} ∪ {x:x>β}
Set notation "and"	α < x < β	{x : α < x < β}
Combined AND	Intersection of both solution sets	Draw number lines, find overlap

Quadratic Inequality Visualiser

Use the sliders to set a, b, c and choose the inequality direction. The parabola plots in real time and the solution region is shaded green.

a = b = c = Direction:

Adjust values to see the solution.

Exercise 1 — Quadratic > 0 (Factorisable)

Each inequality has the form (x−α)(x−β)>0. Enter the SMALLER root (α) as your answer.

1. x²−3x+2>0. Roots are 1 and 2. Enter smaller root.

2. x²−5x+4>0. Roots: 1 and 4. Enter smaller root.

3. x²−7x+12>0. Roots: 3 and 4. Enter smaller root.

4. x²−x−6>0. Roots: −2 and 3. Enter smaller root.

5. x²+x−12>0. Roots: −4 and 3. Enter smaller root.

6. x²−8x+15>0. Roots: 3 and 5. Enter smaller root.

7. x²+2x−8>0. Roots: −4 and 2. Enter smaller root.

8. x²−9>0 (x²−9=(x−3)(x+3)). Enter smaller root.

Exercise 2 — Quadratic < 0 (Factorisable)

Solution is α < x < β. Enter the LARGER root (β).

1. x²−3x+2<0. Between 1 and 2. Enter larger root.

2. x²−5x+4<0. Between 1 and 4. Enter larger root.

3. x²−7x+12<0. Between 3 and 4. Enter larger root.

4. x²−8x+15<0. Roots 3 and 5. Enter larger root.

5. x²−10x+16<0. (x−2)(x−8). Enter larger root.

6. x²−6x+8<0. Roots 2 and 4. Enter larger root.

7. x²+x−2<0. Roots −2 and 1. Enter larger root.

8. x²−16<0. Roots −4 and 4. Enter larger root.

Exercise 3 — Negative a Cases

Multiply by −1 first. Enter the SMALLER root after multiplying out.

1. −x²+5x−6>0. Multiply by −1: x²−5x+6<0. Roots 2,3. Enter smaller root.

2. −x²+7x−10>0. → x²−7x+10<0. Roots 2,5. Enter smaller root.

3. −x²+6x−5<0. → x²−6x+5>0. Roots 1,5. Enter smaller root.

4. −x²+8x−12>0. → x²−8x+12<0. (x−2)(x−6). Enter smaller root.

5. −x²+4x−4≥0. → x²−4x+4≤0. (x−2)²≤0. Only solution x=2. Enter 2.

6. −x²+9>0. → x²−9<0. (x−3)(x+3)<0. Between −3 and 3. Enter smaller root.

7. −2x²+8x−6>0. Divide by −2 (flip): x²−4x+3<0. Roots 1,3. Enter smaller root.

8. −x²+x+6>0. → x²−x−6<0. (x−3)(x+2). Roots −2,3. Enter smaller root.

Exercise 4 — Integer Solutions

Enter the NUMBER of integer solutions (not the integers themselves).

1. x²−5x+4<0. Solution 1<x<4. Integers: {2,3}. Count?

2. x²−7x+10<0. Solution 2<x<5. Integers: {3,4}. Count?

3. x²−6x+8<0. Solution 2<x<4. Integer: {3}. Count?

4. x²−10x+21<0. (x−3)(x−7). Solution 3<x<7. Integers: {4,5,6}. Count?

5. x²−x−6<0. (x−3)(x+2)<0. Solution −2<x<3. Integers: {−1,0,1,2}. Count?

6. x²−4<0. Solution −2<x<2. Integers: {−1,0,1}. Count?

7. x²−9x+14<0. (x−2)(x−7). 2<x<7. Integers: {3,4,5,6}. Count?

8. n²≤25. −5≤n≤5. Integer values of n? Count integers from −5 to 5 inclusive.

Exercise 5 — Combined Quadratic and Linear Inequalities

For the combined solution, enter the UPPER boundary of the right-hand interval.

1. x²−9>0 AND x<8. Left part: x<−3. Right part: 3<x<8. Upper boundary of right interval?

2. x²−4>0 AND x<5. Right part: 2<x<5. Upper boundary?

3. x²−16>0 AND x<10. Right part: 4<x<10. Upper boundary?

4. x²−25>0 AND x<7. Right part: 5<x<7. Upper boundary?

5. x²−1>0 AND x<6. Right part: 1<x<6. Upper boundary?

6. x²−36>0 AND x<9. Right part: 6<x<9. Upper boundary?

7. x²−x−6>0 AND x<7. Roots −2,3. Right part: 3<x<7. Upper boundary?

8. x²−x−2>0 AND x<5. (x−2)(x+1). Roots −1,2. Right part: 2<x<5. Upper boundary?

Practice — 25 Questions

Enter the smaller root for >0 questions (answer = smaller critical value), and larger root for <0 questions (answer = larger critical value), unless specified.

1. x²−5x+6>0. Roots 2,3. Smaller root?

2. x²−5x+6<0. Roots 2,3. Larger root?

3. x²−8x+7>0. (x−1)(x−7). Smaller root?

4. x²−8x+7<0. Larger root?

5. x²−4>0. Roots −2,2. Smaller root?

6. x²−4<0. Larger root?

7. x²−x−12>0. (x−4)(x+3). Smaller root?

8. x²−x−12<0. Larger root?

9. −x²+4x−3>0. Becomes x²−4x+3<0. Roots 1,3. Larger root?

10. −x²+6x−8>0. x²−6x+8<0. Roots 2,4. Larger root?

11. x²−6x+8<0. Integer solutions between 2 and 4: just {3}. Count?

12. x²−8x+12<0. Roots 2,6. Integers {3,4,5}. Count?

13. n²≤9. −3≤n≤3. Count all integers from −3 to 3 inclusive.

14. x²−9>0 AND x<6. Right part 3<x<6. Upper boundary?

15. x²−25>0 AND x<9. Right part 5<x<9. Upper boundary?

16. x²+2x+5>0. Discriminant 4−20<0, always true. For "all real x" enter 0 (as a code).

17. x²+2x−8>0. (x+4)(x−2). Smaller root?

18. x²+2x−8<0. Larger root?

19. −x²+x+6>0. x²−x−6<0. (x−3)(x+2). Roots −2,3. Larger root?

20. 2x²−8<0. x²<4. Roots −2,2. Larger root?

21. x²+3x−4>0. (x+4)(x−1). Roots −4,1. Smaller root?

22. x²+3x−4<0. Larger root?

23. x²−7x+6>0. (x−1)(x−6). Roots 1,6. Smaller root?

24. x²−7x+6<0. Larger root?

25. x²−11x+24<0. (x−3)(x−8). Roots 3,8. Integers in (3,8): {4,5,6,7}. Count?

Challenge — 12 Questions

Harder problems: rearranging first, discriminant analysis, combined, non-integer roots (enter to 2 d.p.).

1. x²>3x+10. Rearrange: x²−3x−10>0. (x−5)(x+2). Smaller root?

2. x(x−4)<12. x²−4x−12<0. (x−6)(x+2). Larger root?

3. 2x²−5x+2<0. Roots: x=(5±3)/4. Roots 0.5 and 2. Larger root?

4. 3x²−7x+2>0. Roots (7±5)/6: 1/3 and 2. Smaller root to 2 d.p.?

5. x²−4x+1<0. Roots (4±√12)/2 = 2±√3. Smaller root to 2 d.p.? [2−1.732=0.27]

6. x²−4>0 AND x²−9<0. Intersection of x<−2 or x>2, and −3<x<3 → 2<x<3. Upper bound?

7. x²−9>0 AND x<5 AND x>0. Right interval: 3<x<5. Upper bound?

8. −2x²+10x−8>0. Divide by −2, flip: x²−5x+4<0. Roots 1,4. Larger root?

9. x²+bx+9>0 for all real x. Need discriminant≤0: b²−36≤0, so −6≤b≤6. Enter max positive value of b.

10. (x−1)²>16. x²−2x+1>16 → x²−2x−15>0. (x−5)(x+3). Smaller root?

11. x²−7x+12≤0. Roots 3,4 inclusive. Integer solutions: 3 and 4. Count?

12. x²−2x−24>0. (x−6)(x+4). Integer: how many integers satisfy x<−4? Trick question — infinitely many. Enter −1 to indicate no finite count (or for IGCSE: smallest integer satisfying x>6 is 7). Enter 7.

Exam Style — 5 Questions

Cambridge IGCSE style questions. Show full working on paper.

[3 marks] Solve x² − 6x + 8 < 0. Enter the LARGER root of the corresponding equation.

[3 marks] Solve −x² + 5x − 4 > 0. This becomes x²−5x+4<0. Enter the LARGER root.

[4 marks] Find the integer values of x that satisfy x²−7x+10 < 0. Roots: 2 and 5. How many integers are strictly between 2 and 5?

[3 marks] Solve x² − 9 > 0 AND x < 7. For the RIGHT-HAND solution interval (positive side), enter the upper bound.

[4 marks] Show that x² + 3x + 5 > 0 for all real x. Discriminant = 9 − 20 = −11. Since discriminant is negative, enter its value.