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 function $\mathrm{f}$ is defined by $\mathrm{f}(x) = x^{2} + ax + b$ for $x \in \mathbb{R}$ , where $a$ and $b$ are constants.
(a) It is given that $a = -4$ and $b = -9$ .

Find the range of $\mathrm{f}$ .

(3 marks)
(b) It is given instead that $a = -10$ and that one root of the equation $\mathrm{f}(x) = 0$ is four times the other root.

Find the value of $b$ .

(3 marks)
(c) Show that, if $b = 2a - 5$ , the equation $\mathrm{f}(x) = 0$ has two distinct real roots for all values of $a$ .

(3 marks)
Level 4 · 9 marksTry it
2
Figure 1

A volleyball player serves the ball. Figure 1 shows the path of the ball, which travels in a vertical plane until it lands on the floor.

The ball is struck at the point $A$ and lands at the point $B$ . The point $O$ is on the floor, vertically below $A$ .

When the ball is $x$ metres horizontally from $O$ , its height above the floor is $H$ metres. Relative to $O$ , the coordinates of $A$ are $(0,\ 2.4)$ and the coordinates of $B$ are $(16,\ 0)$ , as shown in Figure 1.

The ball is at its highest when $x = 5$

The height of the ball is modelled as a quadratic function of its horizontal distance from $O$ .
(a) Find $H$ in terms of $x$ .

(4 marks)
(b) Give one limitation of the model.

(1 mark)

The top of the net is $2.43$ m above the floor, at a horizontal distance of $9$ m from $O$ .
(c) Use the model to determine whether the ball passes over the net.

(2 marks)
Level 4 · 7 marksTry it
3
In this question you must show detailed reasoning.

By using the substitution $u = x^2 + 3x$ , solve the equation $x^4 + 6x^3 + 10x^2 + 3x - 6 = 0$ , giving your answers in exact form.

(5 marks)
Level 4 · 5 marksTry it
4
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve $C_1$ has equation

$y = \frac{12 - x^{3}}{x^{2}} \qquad x \neq 0$

and the curve $C_2$ has equation

$y = 3x^{2} - x - 16$
(a) Show that, where $C_1$ and $C_2$ intersect,

$3x^{4} - 16x^{2} - 12 = 0$

(2 marks)

The curves $C_1$ and $C_2$ intersect at two points, $A$ and $B$ .
(b) Using algebra, find the exact length of $AB$ , giving your answer as a simplified surd.

(5 marks)
Level 4 · 7 marksTry it
5
The equation

$(k - 2)x^2 + 2kx + (k + 1) = 0$ , where $k$ is a constant,

has two distinct real roots.

Find the range of possible values of $k$ . (6 marks)
Level 4 · 6 marksTry it

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