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
The equation $kx^2 + (k + 3)x + 4 = 0$ , where $k$ is a non-zero constant, has two distinct real roots.
(a) Show that $k$ satisfies $k^2 - 10k + 9 > 0$ .
(b) Hence find the set of possible values of $k$ .
Level 5 · 5 marksTry it
2
The equation $x^2 + (k - 3)x + (3 - 2k) = 0$ has two equal roots. Find the possible values of the constant $k$ .
Level 5 · 4 marksTry it
3
The equation $x^2 + (k + 2)x + (k + 5) = 0$ has equal roots.
(a) Show that $k^2 - 16 = 0$ . (3 marks)
(b) Hence find the possible values of $k$ . (2 marks)
Level 5 · 5 marksTry it
4
The line $y = 2x + c$ is a tangent to the curve $y = x^2 - 3x + 7$ . Find the value of the constant $c$ .
Level 5 · 4 marksTry it
5
The equation

$(k - 1)x^2 - 2kx + (k + 3) = 0$

where $k$ is a constant, has at least one real root.

Find the complete set of possible values of $k$ , fully justifying your answer. (7 marks)
Level 5 · 7 marksTry it

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