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^{3} + x^{2} - 20x = 0
x^{3} - 5 x^{2} - 4 x + 20 = 0
x \left(x + 7\right) \left(x + 8\right) = 9 \left(x + 7\right) \left(x + 8\right)
x^{3} - 4 x^{2} - 45 x = 0
x^{3} + 9 x^{2} + 27 x + 27 = 0
- 64 x^{3} + 48 x^{2} - 12 x + 1
x^{3} - 5 x^{2} - 49 x + 245 = 0
x^{3} + 13 x^{2} + 47 x + 35 = 0
150 x^{3} + 115 x^{2} - 118 x - 56 = 0
x^{3} - 3 x^{2} - 18 x + 40 = 0
x^{3} - 4 x^{2} - 45 x = 0
x^{3} + 9 x^{2} + 27 x + 27 = 0
- 64 x^{3} + 48 x^{2} - 12 x + 1
x^{3} - 5 x^{2} - 49 x + 245 = 0
x^{3} + 13 x^{2} + 47 x + 35 = 0
150 x^{3} + 115 x^{2} - 118 x - 56 = 0
x^{3} - 3 x^{2} - 18 x + 40 = 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?
The cubic P \left( x \right) = x^{3} - 7 x^{2} + 14 x - 8 has a factor of x - 1. Solve for the roots of the cubic.
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.
The expression 3 x^{3} - 4 x^{2} - 5 x + 2 = 0 has a factor of x + 1.
Fully factorise 3 x^{3} - 4 x^{2} - 5 x + 2.
Hence, solve the equation 3 x^{3} - 4 x^{2} - 5 x + 2 = 0.
The polynomial x^{3} - 2 x^{2} - 5 x + 6 has a factor of x - 3.
Find the quadratic factor that is multiplied by x - 3 to get x^{3} - 2 x^{2} - 5 x + 6.
Hence, solve the equation x^{3} - 2 x^{2} - 5 x + 6 = 0.
Consider the equation x^{3} + 4 x^{2} - 39 x - 126 = 0.
Given that one solution of the equation is x = - 7, identify one linear factor of
x^{3} + 4 x^{2} - 39 x - 126.
Find the quadratic factor that is multiplied by x + 7 to get x^{3} + 4 x^{2} - 39 x - 126.
Hence, solve the equation x^{3} + 4 x^{2} - 39 x - 126 = 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?
The volume of a reservoir, in \text{mL}, during the first half of the year can be described by the following equation, where 0 \leq t \leq 6 is time in months:
V \left( t \right) = - t^{3} + 30 t^{2} - 131 t + 350
During which months did the reservoir contain 200 \text{ mL}? (Note for January: t=1)
The population of bacteria in a slow-growing culture t hours after 7 am can be described by the equation P \left( t \right) = 200 + 15 t + t^{3}.
What is the initial number of bacteria at 7 am?
How many hours does it take for the initial number of bacteria to double?
Once the number of bacteria reaches 1350, the experiment is stopped. At what time of the day does this happen?
A retail outlet sells x units of a product each year. The annual cost for this product is given by C = 9 x + 3 x^{2} and the annual gross profit is given by G = 9 - 2 x - 4 x^{2} + x^{3}, where C and G are both in dollars.
Form an expression for P, the net profit after producing and selling x units.
How many units must be sold to make a profit of \$40\,672?
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 176 \text{ cm}^3?
To solve a cubic equation of the form x^{3} + m x = n, we can use the formula: x = \sqrt[3]{\dfrac{n}{2} + \sqrt{\left(\dfrac{n}{2}\right)^{2} + \left(\dfrac{m}{3}\right)^{3}}} - \sqrt[3]{\dfrac{- n}{2} + \sqrt{\left(\dfrac{n}{2}\right)^{2} + \left(\dfrac{m}{3}\right)^{3}}}
Use this formula to find the real solution of x^{3} + 24 x = 56.
Use technology to solve the following cubic equations:
147 x^{3} + 427 x^{2} + 160 x - 84 = 0
50 x^{3} + 155 x^{2} + 152 x + 48 = 0
64 x^{3} + 104 x^{2} + 78 x + 18 = 0
48 x^{3} - 212 x^{2} + 84 x + 281 = - 36
- 576 x^{3} + 272 x^{2} - 41 x - 74 = 76
- 25 x^{3} + 20 x^{2} - 5 x + 85 = - 75