5.05 Cubic equations and inequalities
Solve the following cubic equations:
x^{3} = - 8
x^{3} - 49 x = 0
x^{3} - 125 = 0
- 3 x^{3} = 5 x^{2}
8 x^{3} - 125 = 0
\left(x + 8\right) \left(x + 4\right) \left(1 + x\right) = 0
\left( 5 x - 4\right) \left(x + 3\right) \left(x - 2\right) = 0
512 x^{3} - 125 = 0
729 x^{3} + 8 = 0
x(x-4)(x+5) = 0
(x-5)(x-2)(x+2) = 0
x(x+5)(x-9) = 0
(x+3)^3 = 0
(x-4)(x+7)(x-7) = 0
Consider the equation x^{3} - 512 = 0.
Find a value of x that satisfies x^{3} = 512.
Find a factorisation of x^{3} - 512 as a product of a linear and a quadratic factor.
How many zeros does the quadratic factor have?
Hence, how many solutions does x^{3} - 512 = 0 have?
A cubic function is defined as y = x \left(x - 2\right) \left(x - 1\right). Solve for the roots of the cubic.
What is the double root of the function y = 10 x^{2} - x^{3} ?
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.
Consider the function y = x^{3}.
Sketch the graph of the function.
Hence solve the inequality x^{3} < - 8.
Consider the inequality x^{3} - 2 \leq 6.
Solve the equation x^{3} - 2 = 6.
Complete the following table:
| x | 1 | 2 | 3 |
|---|---|---|---|
| x^3-2 | 6 |
Hence, state the solution to the inequality x^{3} - 2 \leq 6.
Consider the function y = 27 - x^{3}.
Sketch the graph of the function.
Hence solve the inequality 27 - x^{3} \leq 0.
Solve the following inequalities:
Consider the inequality 2 - x^{3} \geq x^{3}.
Sketch the graphs of y = x^{3} and y = 2 - x^{3} on the same number plane.
Hence solve the inequality 2 - x^{3} \geq x^{3}.
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
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
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
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.
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?
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 .
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.
If the volume of the can is represented by V, form an expression for V in terms of r only.
What is the radius of the cylinder if the volume is 250 \pi\text{ cm}^3 ?
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}.
Find an expression for the volume of the box in terms of x, in expanded form.
What is the value of x if the volume is 656 \text{ cm}^3?
