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3. Equations and Inequalities
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iGCSE (2021 Edition)

3.05 Quadratic equations with technology

Lesson
Worksheet
Practice
Worksheet
Quadratic equations using technology
1

Solve for the pronumeral for each of the following quadratic equations:

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 for x for each of the following quadratic equations:

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.

Applications
7

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 to find the times at which the revenue will be zero.

8

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

At what lengths, x, will the excess area be zero?

b

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

9

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.

c

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

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Solve quadratic equations for real roots.

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