Topics
14. Non-Linear Equations
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iGCSE (2021 Edition)

14.04 Solving equations with technology

Lesson
Worksheet
Practice
Worksheet
Quadratic equations
1

Solve the following quadratic equations using technology:

a
4 m \left(m + 5\right) = 0
b
\dfrac{m}{2} \left(m + 5\right) = 0
c
- 25 v^{2} + 64 = 0
d
\dfrac{x^{2}}{16} - 2 = 2
e
81 k^{2} + 8 = 24
f
10 \left(p^{2} - 7\right) = 930
g
4 y^{2} = 100
h
25 y^{2} = 36
i
- 3 k^{2} = - 12
2

Solve the following quadratic equations using technology:

a
\left( 8 x + 9\right)^{2} = 256
b
\left(x - 4\right) \left(x - 2\right) = 0
c
x \left(x + 7\right) = 0
d
\left( 8 x - 5\right) \left( 3 x - 7\right) = 0
e
\left( 3 x + 8\right) \left( 5 x - 7\right) = 0
f
\left( 3 x - 17\right) \left( 2 x + d\right) = 0
g
x^{2} = 294
h
81 x^{2} - 16 = 0
i
\left( 10 x - 9\right)^{2} = 0
j
\left( - 3 + 7 x\right)^{2} = 0
k
x \left( 2 x - 9\right) = 0
l
\left(x - 6\right) \left(x + 7\right) = 0
m
x^{2} = 2
n
\left(x - 3\right)^{2} = 64
o
\left(x + 3\right)^{2} = 49
p
\left(2 - x\right)^{2} = 81
q
\left(x - 6\right)^{2} = 2
r
x^{2} = 25
s
x^{2} = 121
t
x^{2} - 121 = 0
u
x^{2} - 10 = 15
3

Solve the following equations by using technology. Round your answers to one decimal place where necessary.

a

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

b

x^{2} + 2 x - \dfrac{21}{4} = 0

c

4.6 x^{2} + 7.3 x - 3.7 = 0

4

Consider the equation 3 x^{2} = 6.

a

Solve the equation by using technology, giving your answers in exact form.

b

Give the solutions as decimals rounded to the nearest tenth.

5

Consider the equation 0 = - x^{2} + 2 x-1. Use technology to solve the equation for x.

a

Write down how many solutions the equation has.

b

Hence, state how many x-intercepts there are on the graph of the function

y = - x^{2} + 2 x-1
6

Solve \left( 5 x^{2} + 13 x + 6\right) \left( 2 x^{2} + 13 x + 20\right) = 0 using technology.

Power equations
7

Solve the following power equations using technology:

a
30^{n} = \sqrt[3]{30}
b
3^{y} = \dfrac{1}{27}
c
4^{y} = 64
d
7^{x} = 1
e
8^{x} = \dfrac{1}{8^{2}}
f
\left(\sqrt{2}\right)^{k} = 0.5
g
\dfrac{1}{36} = 6^{n}
h
\left(\sqrt{6}\right)^{y} = 36
i
5^{x} = \sqrt[3]{5}
j
8^{x} = \sqrt{8}
k
3^{x} = \sqrt[8]{3}
l
4^{x} = 4^{8}
m
x^{5} = 3^{5}
n
9^{x} = 9^{ - 4 }
o
3^{x} = 3^{\frac{2}{9}}
8

Solve the following power equations using technology:

a
x^{ - 7 } = \dfrac{1}{6^{7}}
b
7 \left(4^{x}\right) = \dfrac{7}{4^{3}}
c
10^{x} = 0.01
d
5^{ 10 x + 33} = 125
e
5^{ - 3 x -1} = 3125
f
\left(2^{2}\right)^{x + 7} = 2^{3}
g
\left(2^{4}\right)^{ 2 x - 10} = 2^{2}
h
27 \left(2^{x}\right) = 6^{x}
i
9^{y} = 27
j
3^{ 5 x - 10} = 1
k
\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}
l
30 \times 2^{x - 6} = 15
m
a^{x + 1} = a^{3} \sqrt{a}
n
25^{x + 1} = 125^{ 3 x - 4}
o
\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}
p
3^{x^{2} - 3 x} = 81
q
\left(\dfrac{8}{3}\right)^{x} = \left(\dfrac{8}{3}\right)^{7}
r
9^{x + 3} = 27^{x}
Polynomial equations
9

Solve the following polynomial equations using technology:

a
x^{3} - 125 = 0
b
3 x^{3} = 5 x^{2}
c
x^{3} = - 8
d
8 x^{3} - 125 = 0
e
\left(x + 8\right) \left(x + 4\right) \left(1 + x\right) = 0
f
\left( 5 x - 4\right) \left(x + 3\right) \left(x - 2\right) = 0
g
x^{3} - 49 x = 0
h
512 x^{3} - 125 = 0
i
x^{3} - 3 x^{2} - 18 x + 40 = 0
j
x^{3} - 4 x^{2} - 45 x = 0
k
x^{3} + 9 x^{2} + 27 x + 27 = 0
l
- 64 x^{3} + 48 x^{2} - 12 x + 1 = 0
m
x^{3} - 5 x^{2} - 49 x + 245 = 0
n
x^{3} + 13 x^{2} + 47 x + 35 = 0
o
150 x^{3} + 115 x^{2} - 118 x - 56 = 0
p
729 x^{3} + 8 = 0
q
x^{3} - 5 x^{2} - 4 x + 20 = 0
r
x \left(x - 5\right) \left(x + 7\right) = 8 \left(x - 5\right) \left(x + 7\right)
10

Solve the following polynomial equations using technology:

a
147 x^{3} + 427 x^{2} + 160 x - 84 = 0
b
50 x^{3} + 155 x^{2} + 152 x + 48 = 0
c
64 x^{3} + 40 x^{2} + 54 x + 18 = 0
d
48 x^{3} - 212 x^{2} + 84 x + 209 = - 36
e
- 576 x^{3} + 272 x^{2} - 41 x + 78 = 76
f
- 25 x^{3} + 50 x^{2} - 19 x - 27 = - 45
g
2x^4+5x^3-14x^2-5x+12=0
h
6x^4+9x^3-186x^2+315=0
i
6x^5-5x^4-43x^3+70x^2-24x=0
j
5x^5+x^4-25x^3-5x^2+20x+4=0
Applications
11

The Widget and Trinket Emporium has released the forecast of its revenue over then next year. The revenue R (in dollars) at any point in time t (in months) is described by the equation R = - \left(t - 12\right)^{2} + 4. Solve the equation - \left(t - 12\right)^{2} + 4 = 0 using technology to find the times at which the revenue will be zero.

12

Neville needs a sheet of paper x \text{ cm} by 13\text{ cm} for an origami giraffe. The local origami supply store only sells square sheets of paper.

The lower portion of the image shows the excess area A of paper that will be left after Neville cuts out the x \text{ cm} by 32 \text{ cm} piece. The excess area, in \text{cm}^2, is given by the equation A = x \left(x - 32\right).

a

Use technology to find the lengths, x, that will have an excess area equal to zero?

b

For what value of x will Neville be able to make an origami giraffe with the least amount of excess paper?

13

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. The sum of the heights of both modules equal 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 using technology.

c

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

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Outcomes

0607C2.11

Use of a graphic display calculator to solve equations, including those which may be unfamiliar.

0607E2.10B

Solution of quadratic equations by using a graphic display calculator.

0607E2.11

Use of a graphic display calculator to solve equations, including those which may be unfamiliar.

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