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Next exam on November 14 – 15, 2026:
All concepts

Quadratic Functions and Equations

二次函数与一元二次方程

The parabola of a quadratic function, the roots of a quadratic equation and root–coefficient relations; the discriminant links the equation, function and inequality views.

5 real-exam questions · 5 sittings34 in mock papers14 practice

Formulas

  • Quadratic Formula

    Formula
    x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}

    Applies when：a ≠ 0; real roots exist when b²−4ac ≥ 0

    a, b, c
    coefficients of ax²+bx+c=0 (a ≠ 0)

  • Discriminant

    Definition
    Δ=b2−4ac\Delta=b^{2}-4ac

  • Vieta's Formulas

    Identity
    x1+x2=−ba,x1x2=cax_{1}+x_{2}=-\frac{b}{a},\quad x_{1}x_{2}=\frac{c}{a}

    Applies when：x₁, x₂ are the roots of ax²+bx+c=0 with a ≠ 0

  • Vertex of a Parabola

    Formula
    y=a(x−h)2+k,h=−b2a, k=4ac−b24ay=a(x-h)^{2}+k,\quad h=-\frac{b}{2a},\ k=\frac{4ac-b^{2}}{4a}

    Applies when：a ≠ 0; vertex (h, k), axis of symmetry x = h

How the real exam tests it

Questions from real CSCA papers that test this concept. Work each one out before you open the answer.

  1. CSCA Math December 2025 Past Paper
    The solution set of x2−5x+4<0x^2-5x+4<0 is ( ).
    • A．(1,4)(1,4)
    • B．(−∞,−4)∪(−1,+∞)(-\infty,-4)\cup(-1,+\infty)
    • C．(−4,−1)(-4,-1)
    • D．(−∞,1)∪(4,+∞)(-\infty,1)\cup(4,+\infty)
    Show answer & solution

    Answer: A

    Answer: A x2−5x+4=(x−1)(x−4)<0x^2-5x+4=(x-1)(x-4)<0, so 1<x<41<x<4. Choose A.
    See it in the full CSCA Math December 2025 Past Paper
  2. CSCA Math April 2026 Past Paper
    Solve the inequality x2−2x−3≥0x ^ { 2 } - 2 x - 3 \geq 0.
    • A．(−1,3)( - 1,3 )
    • B．(−∞,−1]∪[3,+∞)( - \infty , - 1 ] \cup [ 3 , + \infty )
    • C．(−∞,−1)∪(3,+∞)( - \infty , - 1 ) \cup ( 3 , + \infty )
    • D．[−1,3][ - 1,3 ]
    Show answer & solution

    Answer: B

    Factor: (x−3)(x+1)≥0( x - 3 ) ( x + 1 ) \geq 0 with roots x=−1,3x = - 1,3. The parabola opens upward, so the expression is non-negative outside the roots: (−∞,−1]∪[3,+∞)( - \infty , - 1 ] \cup [ 3 , + \infty ).
    Take the full CSCA Math April 2026 Past Paper — timed & graded
  3. CSCA Math June 2026 Past Paper
    The solution set of the quadratic inequality 3(x+4)(x−2)<03(x + 4)(x - 2) < 0 is
    • A．(2,+∞)(2, +\infty)
    • B．(−4,2)(-4, 2)
    • C．(−∞,−4)∪(2,+∞)(-\infty, -4) \cup (2, +\infty)
    • D．(−∞,2)∪(−4,+∞)(-\infty, 2) \cup (-4, +\infty)
    Show answer & solution

    Answer: B

    Dividing both sides by the positive number 3 gives (x+4)(x−2)<0(x + 4)(x - 2) < 0. The corresponding parabola opens upward with zeros at −4-4 and 22; take the part of the graph below the xx-axis. The solution set is (−4,2)(-4, 2). Choose (B).
    Take the full CSCA Math June 2026 Past Paper — timed & graded

Real-exam evidence

Recorded past-paper appearances are listed below; online verification is labelled separately.

5 questions across 5 sittings1 of 5 sittings available to verify free online
  • CSCA Math June 2026 Past Paper1 questionreading page not yet public
  • CSCA Math April 2026 Past Paper1 questionreading page not yet public
  • CSCA Math March 2026 Past Paper1 questionreading page not yet public
  • CSCA Math January 2026 Past Paper1 questionreading page not yet public
  • CSCA Math December 2025 Past Paper1 questionView question 3 and answer free (full paper)

34 questions across 16 mock papers — our own papers, so a coverage signal rather than real-exam evidence

14 related practice questions availablePractise free: FunctionPractise free: Inequality

Sitting evidence comes from the CSCA past papers on record; questions whose original paper is missing are excluded, and sprint papers are deliberately not counted.

Glossary

quadratic equationquadratic functiondiscriminantVieta's formulas

Related concepts

Solving InequalitiesArithmetic SequencesGeometric Sequences