4.06 Further quadratic equations
Solve:
5 x^{2} + 45 x = 0
17 x^{2} = 3 x
81 x^{2} - 49 = 0
4 x^{2} - 32 x + 60 = 0
6 x^{2} + 23 x - 4 = 0
- 2 x^{2} - 3 x + 5 = 0
2 x^{2} - 11 x + 15 = 0
3 x^{2} + 14 x + 8 = 0
2 x^{2} + 11 x + 12 = 0
5 x^{2} - 14 x + 8 = 0
12 - 7 b - 12 b^{2} = 0
3 x^{2} + 5 x + 2 = 0
3 x^{2} - 12 x - 36 = 0
4 x^{2} - 3 x - 10 = 0
4 x^{2} - 13 x + 2 = 0
4 k^{2} = - 15 + 17 k
Solve:
\dfrac{4 x^{2} - 55 x}{5} = 15
\left( 3 x^{2} + 11 x + 10\right) \left( 2 x^{2} + 11 x + 12\right) = 0
\dfrac{2 x}{x^{2} - 12} = \sqrt{2}
\dfrac{4 x + 1}{4 x - 1} - \dfrac{4 x - 1}{4 x + 1} = 7
The equation 4 x^{2} + k x + 36 = 0 has one solution x = - 3. Find the value of the coefficient k.
Consider the quadratic expression a x^{2} - 6 x + 144.
If you substitute x = 6 into the expression, what must be the value of a to make the expression equal 0?
Consider the graph of the parabola:
State the x-values of the x-intercepts of the parabola.
State the y-value of the y-intercept for this curve.
State the equation of the vertical axis of symmetry for this parabola.
What are the coordinates of the parabola's vertex?
Determine whether the following statements are true about this vertex:
The y-value of the vertex is the same as the y-value of the y-intercept.
The vertex is the maximum value of the graph.
The vertex is the minimum value of the graph.
The vertex lies on the axis of symmetry.
The x-value of the vertex is the average of the x-values of the two x-intercepts.
Consider the graph of the parabola below:
Find the x-values of the x-intercepts of the parabola.
Find the y-value of the y-intercept for this curve.
Find the equation of the vertical axis of symmetry for this parabola.
What are the coordinates of the parabola's vertex?
Determine whether the following statements are true about this vertex:
The vertex is the maximum value of the graph.
The vertex is the minimum value of the graph.
The x-value of the vertex is the average of the x-values of the two x-intercepts.
The vertex lies on the axis of symmetry.
The y-value of the vertex is the same as the y-value of the y-intercept.
Consider the graph of the parabola below:
Find the x-values of the x-intercepts of the parabola.
Find the y-value of the y-intercept for this curve.
Find the equation of the vertical axis of symmetry for this parabola.
What are the coordinates of the parabola's vertex?
Determine whether the following statements are true about this vertex:
The vertex is the maximum value of the graph.
The y-value of the vertex is the same as the y-value of the y-intercept.
The x-value of the vertex is the average of the x-values of the two x-intercepts.
The vertex is the minimum value of the graph.
The vertex lies on the axis of symmetry.
A parabola is described by y = 9 x^{2} - 54 x + 72.
Find the x-values of the x-intercepts of the parabola.
Express the equation y = 9 x^{2} - 54 x + 72 in the form y = a \left(x - h\right)^{2} + k.
Find the coordinates of the vertex for this parabola.
Hence, sketch the graph of the parabola.
A parabola is described by y = - 2 x^{2} + 12 x - 10.
Find the x-values of the x-intercepts of the parabola.
Find the y-value of the y-intercept for this curve.
Find the vertical axis of symmetry for this parabola.
Find the y-coordinate of the vertex of the parabola.
Hence, sketch the graph of the parabola.
A parabola is described by y = x^{2} + 2 x.
Find the x-values of the x-intercepts of the parabola.
Find the y-value of the y-intercept for this parabola.
Find the equation of the axis of symmetry for this parabola.
Find the coordinates of the vertex for this parabola.
Hence, sketch the graph of the parabola.
Consider the parabola defined by y = 2 x^{2} - 8.
Find the x-values of the x-intercepts of the parabola.
Find the vertical axis of symmetry for this parabola.
Find the coordinates of the vertex for this parabola.
Hence, sketch the graph of the parabola.
A parabola is of the form y = \left(x - h\right)^{2} + k. It has x-intercepts at \left(9, 0\right) and \left(10, 0\right).
Determine the axis of symmetry of the curve.
Hence, find the equation of the curve in the form y = \left(x - h\right)^{2} + k.
The height at time t of a ball thrown upwards is given by the equation h = 59 + 40 t - 5 t^{2}.
How long does it take the ball to reach its maximum height?
Find the height of the ball at its highest point.
Mae throws a stick vertically upwards. After t seconds, its height h metres above the ground is given by the formula h = 25 t - 5 t^{2}.
At what time(s) will the stick be 30 \text{ m} above the ground?
How long does the stick take to hit the ground?
Can the stick ever reach a height of 36 \text{ m}?
A t-shirt is fired straight up from a t-shirt cannon at ground level. After t seconds, its height above the ground is h feet, where h = - 16 t^{2} + 48 t.
For what values of t is the t-shirt 11 \text{ ft} above the ground?
Hence, calculate how long the t-shirt was at least 11 \text{ ft} above the ground.
For what values of t is the t-shirt 35 \text{ ft} above the ground?
Hence, calculate how long the t-shirt was at least 35 \text{ ft} above the ground.
The Widget and Trinket Emporium has released the forecast of its revenue over the next year. The revenue R (in dollars) at any point in time t (in months) is described by the equation
R = - \left(2t - 18\right)^{2} + 16Find the times at which the revenue will be zero.
An interplanetary freight transport company has won a contract to supply the space station orbiting Mars. They will be shipping stackable containers, each carrying a fuel module and a water module, that must meet certain dimension restrictions.
The design engineers have produced a sketch for the modules and container, shown below. Each component has a square cross-sectional area of x^{2} \text{ cm}^2, and the heights of both modules sum to the height of the container.
Write an equation that equates the height of the container and the sum of the heights of the modules.
Find the possible values of x.
Find the tallest possible height of the container. Give your answer to two decimal places.