Quadratic Equations and Inequalities

Maths, SSC-Punjab Class 10. This chapter is split into 42 topics and has 1,086 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Changing the subject of a formula: basic rearrangements 26 questions
2. Zero product property and checking roots 26 questions
3. Factorization method: non-monic quadratic equations 29 questions
4. Applications: quadratic inequalities in context 24 questions
5. Quadratic formula: equations with rational roots 29 questions
6. Fractional equations solved with the quadratic formula 24 questions
7. Graph of a quadratic function: shape and vertex form 27 questions
8. Quadratic equation: definition and standard form 26 questions
9. Quadratic inequalities with equality, closed intervals and notation 26 questions
10. Changing the subject of a formula: squares, roots and fractions 26 questions
11. Values of a constant for which the roots are real, unequal or imaginary 24 questions
12. Completing the square: solving equations 27 questions
13. Equations from fractional and irrational roots 26 questions
14. Solving quadratic equations graphically 26 questions
15. Derivation of the quadratic formula 23 questions
16. Nature of the roots: rules and classification 28 questions
17. Solving quadratic inequalities: the four steps 26 questions
18. Sum and product of the roots 26 questions
19. Factorization method: monic quadratic equations 30 questions
20. Symmetric functions of the roots 26 questions
21. Coefficients a, b, c and types of quadratic equations 26 questions
22. Special relations between the roots and the coefficients 24 questions
23. Finding unknown constants from conditions on the roots 26 questions
24. Applications: maximum and minimum values (profit and earnings) 24 questions
25. Completing the square: the procedure 27 questions
26. Choosing among the methods of solving quadratic equations 24 questions
27. Equations whose roots are functions of the roots of a given equation 26 questions
28. Fractional equations reduced to quadratic equations 27 questions
29. Intersection of a line and a parabola 26 questions
30. Condition for equal roots: finding the unknown constant 24 questions
31. Review exercise of Unit 2 24 questions
32. Factorization: equations needing rearrangement 27 questions
33. Applications: motion of a ball thrown upward 24 questions
34. Quadratic formula: equations with imaginary roots 25 questions
35. Intersection of a linear function with the coordinate axes 26 questions
36. Vertex, axis of symmetry and intercepts of a parabola 28 questions
37. Intersection of two linear functions 26 questions
38. The discriminant of a quadratic equation 26 questions
39. Quadratic inequalities: perfect squares, no real roots and surd roots 24 questions
40. Applications: word problems leading to quadratic equations 24 questions
41. Quadratic formula: equations with irrational roots 27 questions
42. Forming a quadratic equation from given roots 26 questions

Sample questions: Quadratic Equations and Inequalities

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Which is the correct order of the steps in deriving the quadratic formula?

• A. divide by a; take square roots; add (b/2a)²; transpose c; transpose b/2a
• B. add (b/2a)²; transpose c; divide by a; take square roots; transpose b/2a
• C. transpose c; divide by a; add (b/2a)²; take square roots; transpose b/2a(correct answer)
• D. take square roots; transpose c; divide by a; add (b/2a)²; transpose b/2a

Answer: C.Steps I to IV are transpose c, divide by a, add the square of half the coefficient of x, take square roots, and finally transpose b/2a.

Q2. Which statement is true for the roots obtained by factorization, completing the square and the formula?

• A. All three methods give the same roots(correct answer)
• B. The formula gives only rational roots
• C. Factorization gives different roots from the formula
• D. Completing the square gives only the larger root

Answer: A.Each method solves the same equation, so the roots are the same. Review Exercise 2, Q2 asks the student to use all three on the same equations.

Q3. Challenge: one root of a quadratic is twice the other and their product is 8. The equation (monic) is

• A. x² − 2x + 8 = 0
• B. x² − 6x + 8 = 0 or x² + 6x + 8 = 0(correct answer)
• C. x² − 3x + 8 = 0
• D. x² − 6x + 16 = 0

Answer: B.Roots r and 2r with 2r² = 8 give r = ±2, so the roots are 2, 4 or −2, −4 and the equations are x² − 6x + 8 = 0 or x² + 6x + 8 = 0.

Q4. Review Exercise 2(viii): if b² − 4ac > 0 and is a perfect square, the roots of ax² + bx + c = 0 are

• A. equal
• B. imaginary
• C. irrational
• D. unequal(correct answer)

Answer: D.A positive discriminant gives two different real roots; with a perfect square they are rational, so (d) irrational is false and (b) unequal is the correct choice.

Q5. Example 17: the solution of 2x² − 5x + 2 ≤ 0 is

• A. (−∞, 1/2) ∪ (2, ∞)
• B. (1/2, 2)
• C. [1/2, 2](correct answer)
• D. (−∞, 1/2] ∪ [2, ∞)

Answer: C.Test points: x = 0 gives 2 > 0 (no), x = 1 gives −1 < 0 (yes), x = 3 gives 5 > 0 (no); the endpoints are included for ≤, so [1/2, 2].

Q6. Solve −3x² − 18x − 15 > 0.

• A. (−5, −1)(correct answer)
• B. [−5, −1]
• C. (−∞, −5] ∪ [−1, ∞)
• D. (−∞, −5) ∪ (−1, ∞)

Answer: A.Here a = −3 is negative, so the parabola opens downward: the quadratic is positive between the roots −5 and −1 and negative outside them.

Q7. Solve −x² − 5x + 24 ≤ 0.

• A. (−∞, −8) ∪ (3, ∞)
• B. (−8, 3)
• C. [−8, 3]
• D. (−∞, −8] ∪ [3, ∞)(correct answer)

Answer: D.Here a = −1 is negative, so the parabola opens downward: the quadratic is positive between the roots −8 and 3 and negative outside them.

Q8. Solve −x² − 16x − 63 > 0.

• A. (−∞, −9) ∪ (−7, ∞)
• B. (−9, −7)(correct answer)
• C. [−9, −7]
• D. (−∞, −9] ∪ [−7, ∞)

Answer: B.Here a = −1 is negative, so the parabola opens downward: the quadratic is positive between the roots −9 and −7 and negative outside them.

Q9. Solve −3x² − 30x − 48 ≤ 0.

• A. [−8, −2]
• B. (−∞, −8) ∪ (−2, ∞)
• C. (−∞, −8] ∪ [−2, ∞)(correct answer)
• D. (−8, −2)

Answer: C.Here a = −3 is negative, so the parabola opens downward: the quadratic is positive between the roots −8 and −2 and negative outside them.

Q10. A formula is

• A. a number written without any variables or symbols
• B. an equation expressing a precise relationship between measurable quantities(correct answer)
• C. a graph that shows the roots of a quadratic equation
• D. an inequality that compares two measurable numbers only

Answer: B.Section 2.6 defines a formula as expressing the precise relationship between various measurable quantities in a problem or situation in an equation form.