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Topics · Quadratic Equations

Quadratic equations in DSE Maths Paper 2

Quadratics turn up in both sections of Paper 2. You need to solve quadratic equations, use the discriminant to decide how many real roots there are, use the sum and product of roots without solving, and find the greatest or least value of a quadratic function from its graph or by completing the square.

What this topic covers

  • Solving quadratic equations by factorising, the formula or completing the square
  • The discriminant and the nature of the roots
  • Sum and product of roots, and expressions built from them
  • Greatest and least values; the vertex and axis of symmetry of the graph

Read the study guide on this topic

Two sample questions

Sample 1

If y equals negative x squared plus 10x minus 3, where x is a real number, then the greatest value of y is

  1. A22.
  2. B28.
  3. C47.
  4. D97.
Show the answer and solution

Answer: A

Worked solution

y=−x2+10x−3\displaystyle y = -x^2 + 10x - 3=−(x−5)2+22\displaystyle = -(x - 5)^2 + 22Since −(x−5)2≤0\displaystyle -(x - 5)^2 \leq 0 for every real x\displaystyle x, the greatest value of y\displaystyle y is 22.\displaystyle 22.

Sample 2

Let alpha and beta be the roots of the equation x squared plus 3kx minus 4 equals 0, where k is a constant. Then 1 over alpha squared plus 1 over beta squared equals

  1. A16 minus 6k over 9k squared .
  2. B9k squared plus 8 over 16 .
  3. C16 plus 6k over 9k squared .
  4. D9k squared minus 8 over 16 .
Show the answer and solution

Answer: B

Worked solution

α+β=−3k,αβ=−4\displaystyle \alpha + \beta = -3k, \quad \alpha\beta = -41α2\displaystyle \dfrac{1}{\alpha^2}+1β2\displaystyle {}+ \dfrac{1}{\beta^2}=α2+β2(αβ)2\displaystyle {}= \dfrac{\alpha^2 + \beta^2}{(\alpha\beta)^2}=(α+β)2−2αβ(αβ)2\displaystyle {}= \dfrac{(\alpha + \beta)^2 - 2\alpha\beta}{(\alpha\beta)^2}=(−3k)2−2(−4)(−4)2\displaystyle = \dfrac{(-3k)^2 - 2(-4)}{(-4)^2}=9k2+816\displaystyle {}= \dfrac{9k^2 + 8}{16}

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Original material by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.