A-level · Algebra & functions

Quadratic equationsExam-style questions

Solving a quadratic equation and reading what the answer says. Why a quadratic has two, one or no real roots; solving by factorising, and the root lost by dividing by x; completing the square to find the turning point; the quadratic formula, where it comes from and why the whole top is over 2a; the discriminant, which counts the roots before you solve; and conditions on an unknown constant, including a line that touches a curve.

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25 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Write $3x^2 - 18x + 31$ in the form $3(x - a)^2 + b$ , where $a$ and $b$ are constants to be determined.

(2 marks)
(b) Use your answer to part (a) to state the minimum value of $3x^2 - 18x + 31$ .

(1 mark)
Level 2 · 3 marksTry it
2
Solve $x^4 - 5x^2 + 4 = 0$ . (5 marks)
Level 3 · 5 marksTry it
3
(a) Express $2y^{2} - 12y + 9$ in the form $2(y + a)^{2} + b$ , where $a$ and $b$ are constants.

(2 marks)
(b) Hence find the exact solutions of the equation $2x - 12\sqrt{x} + 9 = 0$ .

(3 marks)
Level 3 · 5 marksTry it
4
Determine the set of values of $k$ such that the equation $(x - 3)^2 + 2k - 5 = 0$ has two distinct real roots.

(4 marks)
Level 3 · 4 marksTry it
5
Determine the set of values of $k$ such that the equation $2x^2 + (k - 1)x + 8 = 0$ has two distinct real roots. Give your answer in set notation.

(4 marks)
Level 3 · 4 marksTry it

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