topic badge
Australia
10&10a

4.06 Further quadratic equations

Worksheet
Further quadratic equations
1

Solve:

a

5 x^{2} + 45 x = 0

b

17 x^{2} = 3 x

c

81 x^{2} - 49 = 0

d

4 x^{2} - 32 x + 60 = 0

e

6 x^{2} + 23 x - 4 = 0

f

- 2 x^{2} - 3 x + 5 = 0

g

2 x^{2} - 11 x + 15 = 0

h

3 x^{2} + 14 x + 8 = 0

i

2 x^{2} + 11 x + 12 = 0

j

5 x^{2} - 14 x + 8 = 0

k

12 - 7 b - 12 b^{2} = 0

l

3 x^{2} + 5 x + 2 = 0

m

3 x^{2} - 12 x - 36 = 0

n

4 x^{2} - 3 x - 10 = 0

o

4 x^{2} - 13 x + 2 = 0

p

4 k^{2} = - 15 + 17 k

2

Solve:

a

\dfrac{4 x^{2} - 55 x}{5} = 15

b

\left( 3 x^{2} + 11 x + 10\right) \left( 2 x^{2} + 11 x + 12\right) = 0

c

\dfrac{2 x}{x^{2} - 12} = \sqrt{2}

d

\dfrac{4 x + 1}{4 x - 1} - \dfrac{4 x - 1}{4 x + 1} = 7

3

The equation 4 x^{2} + k x + 36 = 0 has one solution x = - 3. Find the value of the coefficient k.

4

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?

Parabolas
5

Consider the graph of the parabola:

-6
-4
-2
2
4
6
x
-16
-14
-12
-10
-8
-6
-4
-2
2
y
a

State the x-values of the x-intercepts of the parabola.

b

State the y-value of the y-intercept for this curve.

c

State the equation of the vertical axis of symmetry for this parabola.

d

What are the coordinates of the parabola's vertex?

e

Determine whether the following statements are true about this vertex:

i

The y-value of the vertex is the same as the y-value of the y-intercept.

ii

The vertex is the maximum value of the graph.

iii

The vertex is the minimum value of the graph.

iv

The vertex lies on the axis of symmetry.

v

The x-value of the vertex is the average of the x-values of the two x-intercepts.

6

Consider the graph of the parabola below:

-2
-1
1
2
3
4
5
6
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
y
a

Find the x-values of the x-intercepts of the parabola.

b

Find the y-value of the y-intercept for this curve.

c

Find the equation of the vertical axis of symmetry for this parabola.

d

What are the coordinates of the parabola's vertex?

e

Determine whether the following statements are true about this vertex:

i

The vertex is the maximum value of the graph.

ii

The vertex is the minimum value of the graph.

iii

The x-value of the vertex is the average of the x-values of the two x-intercepts.

iv

The vertex lies on the axis of symmetry.

v

The y-value of the vertex is the same as the y-value of the y-intercept.

7

Consider the graph of the parabola below:

-4
-2
2
4
6
8
x
-5
5
10
15
20
25
y
a

Find the x-values of the x-intercepts of the parabola.

b

Find the y-value of the y-intercept for this curve.

c

Find the equation of the vertical axis of symmetry for this parabola.

d

What are the coordinates of the parabola's vertex?

e

Determine whether the following statements are true about this vertex:

i

The vertex is the maximum value of the graph.

ii

The y-value of the vertex is the same as the y-value of the y-intercept.

iii

The x-value of the vertex is the average of the x-values of the two x-intercepts.

iv

The vertex is the minimum value of the graph.

v

The vertex lies on the axis of symmetry.

8

A parabola is described by y = 9 x^{2} - 54 x + 72.

a

Find the x-values of the x-intercepts of the parabola.

b

Express the equation y = 9 x^{2} - 54 x + 72 in the form y = a \left(x - h\right)^{2} + k.

c

Find the coordinates of the vertex for this parabola.

d

Hence, sketch the graph of the parabola.

9

A parabola is described by y = - 2 x^{2} + 12 x - 10.

a

Find the x-values of the x-intercepts of the parabola.

b

Find the y-value of the y-intercept for this curve.

c

Find the vertical axis of symmetry for this parabola.

d

Find the y-coordinate of the vertex of the parabola.

e

Hence, sketch the graph of the parabola.

10

A parabola is described by y = x^{2} + 2 x.

a

Find the x-values of the x-intercepts of the parabola.

b

Find the y-value of the y-intercept for this parabola.

c

Find the equation of the axis of symmetry for this parabola.

d

Find the coordinates of the vertex for this parabola.

e

Hence, sketch the graph of the parabola.

11

Consider the parabola defined by y = 2 x^{2} - 8.

a

Find the x-values of the x-intercepts of the parabola.

b

Find the vertical axis of symmetry for this parabola.

c

Find the coordinates of the vertex for this parabola.

d

Hence, sketch the graph of the parabola.

12

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).

a

Determine the axis of symmetry of the curve.

b

Hence, find the equation of the curve in the form y = \left(x - h\right)^{2} + k.

Applications
13

The height at time t of a ball thrown upwards is given by the equation h = 59 + 40 t - 5 t^{2}.

a

How long does it take the ball to reach its maximum height?

b

Find the height of the ball at its highest point.

14

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}.

a

At what time(s) will the stick be 30 \text{ m} above the ground?

b

How long does the stick take to hit the ground?

c

Can the stick ever reach a height of 36 \text{ m}?

15

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.

a

For what values of t is the t-shirt 11 \text{ ft} above the ground?

b

Hence, calculate how long the t-shirt was at least 11 \text{ ft} above the ground.

c

For what values of t is the t-shirt 35 \text{ ft} above the ground?

d

Hence, calculate how long the t-shirt was at least 35 \text{ ft} above the ground.

16

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} + 16

Find the times at which the revenue will be zero.

17

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.

a

Write an equation that equates the height of the container and the sum of the heights of the modules.

b

Find the possible values of x.

c

Find the tallest possible height of the container. Give your answer to two decimal places.

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

ACMNA269 (10a)

Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts

What is Mathspace

About Mathspace