Theory of Quadratic Functions

Maths, HSSC-Punjab First Year. This chapter is split into 22 topics and has 505 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

  1. Checking solutions and extraneous roots 21 questions
  2. Absolute value inequalities: greater than 21 questions
  3. Intercepts of a quadratic function 25 questions
  4. Maximum or minimum point, domain and range 23 questions
  5. Quadratic formula, discriminant and roots 27 questions
  6. Radical equations reducible to quadratic equations 25 questions
  7. Word problems: dimensions and numbers 22 questions
  8. Absolute value quadratic equations 24 questions
  9. Quadratic function and its standard form 24 questions
  10. Domain and range of inverse functions 24 questions
  11. Maximum and minimum values by completing the square 26 questions
  12. Absolute value 23 questions
  13. Inverse of a quadratic function: restricted domain 20 questions
  14. Increasing and decreasing intervals 22 questions
  15. Shape of the parabola and axis of symmetry 24 questions
  16. Vertex form of a quadratic function 23 questions
  17. Rational equations reducible to quadratic equations 23 questions
  18. Absolute value inequalities: less than 21 questions
  19. Word problems: prices, speeds and work 20 questions
  20. Formulas for inverses of quadratic functions 21 questions
  21. Vertex of a parabola 24 questions
  22. Quadratic models: projectile, braking distance and profit 22 questions

Sample questions: Theory of Quadratic Functions

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

Q1. The solution set of |x² + 3x − 4| > 2 is

  • A. (−3/2 − √33/2, −3/2 − √17/2) ∪ (−3/2 + √17/2, −3/2 + √33/2)
  • B. (−∞, −3/2 − √33/2] ∪ [−3/2 − √17/2, −3/2 + √17/2] ∪ [−3/2 + √33/2, ∞)
  • C. (−∞, −3/2 − √33/2) ∪ (−3/2 − √17/2, −3/2 + √17/2) ∪ (−3/2 + √33/2, ∞)
  • D. (−∞, −3/2 − √33/2) ∪ (−3/2 + √33/2, ∞)

Answer: C.The inequality splits into x² + 3x − 4 > 2 or x² + 3x − 4 < −2; the union is (−∞, −3/2 − √33/2) ∪ (−3/2 − √17/2, −3/2 + √17/2) ∪ (−3/2 + √33/2, ∞).

Q2. The solution set of |x² + 7x − 2| ≥ 4 is

  • A. (−∞, −7/2 − √73/2) ∪ (−7/2 − √41/2, −7/2 + √41/2) ∪ (−7/2 + √73/2, ∞)
  • B. [−7/2 − √73/2, −7/2 − √41/2] ∪ [−7/2 + √41/2, −7/2 + √73/2]
  • C. (−∞, −7/2 − √73/2) ∪ (−7/2 + √73/2, ∞)
  • D. (−∞, −7/2 − √73/2] ∪ [−7/2 − √41/2, −7/2 + √41/2] ∪ [−7/2 + √73/2, ∞)

Answer: D.The inequality splits into x² + 7x − 2 ≥ 4 or x² + 7x − 2 ≤ −4; the union is (−∞, −7/2 − √73/2] ∪ [−7/2 − √41/2, −7/2 + √41/2] ∪ [−7/2 + √73/2, ∞).

Q3. The solution set of |x² + x − 2| > 1 is

  • A. (−∞, −1/2 − √13/2) ∪ (−1/2 − √5/2, −1/2 + √5/2) ∪ (−1/2 + √13/2, ∞)
  • B. (−∞, −1/2 − √13/2] ∪ [−1/2 − √5/2, −1/2 + √5/2] ∪ [−1/2 + √13/2, ∞)
  • C. (−∞, −1/2 − √13/2) ∪ (−1/2 + √13/2, ∞)
  • D. (−1/2 − √13/2, −1/2 − √5/2) ∪ (−1/2 + √5/2, −1/2 + √13/2)

Answer: A.The inequality splits into x² + x − 2 > 1 or x² + x − 2 < −1; the union is (−∞, −1/2 − √13/2) ∪ (−1/2 − √5/2, −1/2 + √5/2) ∪ (−1/2 + √13/2, ∞).

Q4. The real solutions of 4/x + 2/(x + 3) = 3 are

  • A. {1/8 − √17/8, 1/8 + √17/8}
  • B. {1/2 − √17/2, 1/2 + √17/2}
  • C. {−1/2 − √17/2, 0, −1/2 + √17/2}
  • D. {−1/2 − √17/2, −1/2 + √17/2}

Answer: D.Multiply by x(x + 3) to get −3x² − 3x + 12 = 0; the excluded values 0 and −3 are not roots, so the solutions are {−1/2 − √17/2, −1/2 + √17/2}.

Q5. The real solutions of 4/x + 5/(x + 3) = 4 are

  • A. {3/8 − √201/8, 3/8 + √201/8}
  • B. {−3/8 − √201/8, −3/8 + √201/8}
  • C. {5/8 − √201/8, 5/8 + √201/8}
  • D. {−3/8 − √201/8, 0, −3/8 + √201/8}

Answer: B.Multiply by x(x + 3) to get −4x² − 3x + 12 = 0; the excluded values 0 and −3 are not roots, so the solutions are {−3/8 − √201/8, −3/8 + √201/8}.

Q6. The real solutions of 4/x + 3/(x + 3) = 2 are

  • A. {1/4 − √97/4, 1/4 + √97/4}
  • B. {−1/4 − √97/4, −1/4 + √97/4}
  • C. {1/4 − √97/4, 0, 1/4 + √97/4}
  • D. {5/4 − √97/4, 5/4 + √97/4}

Answer: A.Multiply by x(x + 3) to get −2x² + x + 12 = 0; the excluded values 0 and −3 are not roots, so the solutions are {1/4 − √97/4, 1/4 + √97/4}.

Q7. The real solutions of 4/x + 1/(x + 3) = 4 are

  • A. {1/8 − √241/8, 1/8 + √241/8}
  • B. {−7/8 − √241/8, −7/8 + √241/8}
  • C. {7/8 − √241/8, 7/8 + √241/8}
  • D. {−7/8 − √241/8, 0, −7/8 + √241/8}

Answer: B.Multiply by x(x + 3) to get −4x² − 7x + 12 = 0; the excluded values 0 and −3 are not roots, so the solutions are {−7/8 − √241/8, −7/8 + √241/8}.

Q8. The real solutions of 4/x + 2/(x + 2) = 2 are

  • A. {3/2 − √17/2, 3/2 + √17/2}
  • B. {1/2 − √17/2, 1/2 + √17/2}
  • C. {1/2 − √17/2, 0, 1/2 + √17/2}
  • D. {−1/2 − √17/2, −1/2 + √17/2}

Answer: B.Multiply by x(x + 2) to get −2x² + 2x + 8 = 0; the excluded values 0 and −2 are not roots, so the solutions are {1/2 − √17/2, 1/2 + √17/2}.

Q9. Extraneous roots of a radical equation are

  • A. roots that are lost on squaring both sides
  • B. roots that are always imaginary numbers
  • C. roots of the radical-free equation that fail the original
  • D. roots that satisfy the original equation twice over

Answer: C.The book: the new equation has solutions that are not solutions of the original radical equation; such extra roots are called extraneous roots.

Q10. The solution set of √(3x + 7) = x − 5 is

  • A. {13/2 − √97/2}
  • B. {13/2 − √97/2, 13/2 + √97/2}
  • C. {13/2 + √97/2}
  • D. {−13/2 − √97/2}

Answer: C.Squaring gives {13/2 − √97/2, 13/2 + √97/2}; checking in the original equation keeps only {13/2 + √97/2} (the others make x − 5 negative).

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