iGCSE (2021 Edition)

# 5.05 Cubic equations and inequalities

Worksheet
Solving cubic equations
1

Solve the following cubic equations:

a

x^{3} = - 8

b

x^{3} - 49 x = 0

c

x^{3} - 125 = 0

d

- 3 x^{3} = 5 x^{2}

e

8 x^{3} - 125 = 0

f

\left(x + 8\right) \left(x + 4\right) \left(1 + x\right) = 0

g

\left( 5 x - 4\right) \left(x + 3\right) \left(x - 2\right) = 0

h

512 x^{3} - 125 = 0

i

729 x^{3} + 8 = 0

j

x(x-4)(x+5) = 0

k

(x-5)(x-2)(x+2) = 0

l

x(x+5)(x-9) = 0

m

(x+3)^3 = 0

n

(x-4)(x+7)(x-7) = 0

2

Consider the equation x^{3} - 512 = 0.

a

Find a value of x that satisfies x^{3} = 512.

b

Find a factorisation of x^{3} - 512 as a product of a linear and a quadratic factor.

c

How many zeros does the quadratic factor have?

d

Hence, how many solutions does x^{3} - 512 = 0 have?

3

A cubic function is defined as y = x \left(x - 2\right) \left(x - 1\right). Solve for the roots of the cubic.

4

What is the double root of the function y = 10 x^{2} - x^{3} ?

5

One of the solutions of the equation \left( a x + 4\right) \left(x + 6\right) \left(x - 7\right) = 0 is x = - 4.

Solve for the value of a.

Cubic inequalities
6

Consider the function y = x^{3}.

a

Sketch the graph of the function.

b

Hence solve the inequality x^{3} < - 8.

7

Consider the inequality x^{3} - 2 \leq 6.

a

Solve the equation x^{3} - 2 = 6.

b

Complete the following table:

c

Hence, state the solution to the inequality x^{3} - 2 \leq 6.

8

Consider the function y = 27 - x^{3}.

a

Sketch the graph of the function.

b

Hence solve the inequality 27 - x^{3} \leq 0.

9

Solve the following inequalities:

a
x^{3} \geq 125
b
x^{3} + 3 > 0
c
\dfrac{x^{3}}{2} + 2 > 0
d
8 x^{3} \leq 125
e
81 - 3 x^{3} > 0
f
\left(x + 2\right)^{3} > - 64
g
- 27 \left(x - 3\right)^{3} \leq - 8
10

Consider the inequality 2 - x^{3} \geq x^{3}.

a

Sketch the graphs of y = x^{3} and y = 2 - x^{3} on the same number plane.

b

Hence solve the inequality 2 - x^{3} \geq x^{3}.

11

Use the following graph of the function \\ y = - \left(x + 2\right)^{2} \left(x - 2\right) to solve the inequality: - \left(x + 2\right)^{2} \left(x - 2\right) \leq 0

12

Use the following graph of the function \\ y = - \left(x - 1\right)^{2} \left(x - 3\right) to solve the inequality: \left(x - 1\right)^{2} \left(x - 3\right) \leq 0

13

Use the following graph of the function \\ y = \left(x - 1\right) \left(x + 1\right) \left(x - 3\right) to solve the inequality: \left(x - 1\right) \left(x + 1\right) \left(x - 3\right) \leq 0

Applications
14

The volume of a sphere is given by the formula V = \dfrac{4}{3} \pi r^{3}.

If a sphere of radius r\text{ m} has volume 288 \pi\text{ m}^3, find the value of r.

15

The mass (M) in kilograms of a cubic container of water is given by M = 0.001 r^{3}, where r is the side length of the cube-shaped container in centimetres.

The greatest mass Ivan can carry is equivalent to his own weight, which is 73 kilograms. What is the length (in \text{cm}) of the largest cubic container of water Ivan can carry correct to two decimal places?

16

A cylindrical can is to be designed using a fixed amount of tin material, so that the total surface area including the top and bottom faces is 150 \pi \text{ cm}^2 .

a

If the radius of the cylinder is represented by r, and the height of the cylinder is represented by h, express h in terms of r.

b

If the volume of the can is represented by V, form an expression for V in terms of r only.

c

What is the radius of the cylinder if the volume is 250 \pi\text{ cm}^3 ?

17

A box is formed by cutting squares of length x\text{ cm} from the corners of a piece of cardboard 10\text{ cm} by 30\text{ cm}.

a

Find an expression for the volume of the box in terms of x, in expanded form.

b

What is the value of x if the volume is 656 \text{ cm}^3?

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### Outcomes

#### 0606C3.5

Solve cubic inequalities in the form k(x – a)(x – b)(x – c) ⩽ d graphically.

#### 0606C5.3

Solve cubic equations.