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Middle Years

2.05 Quadratic formula

Worksheet
The quadratic formula
1

Considering the quadratic formula, find the values of a, b and c in the following quadratic equations:

a
x^{2} + 7 x + 10 = 0
b
x^{2} - 3 x - 4 = 0
c
2 x^{2} + 9 x = 0
d
4 x^{2} + 3 x = 5
e
8 x^{2} - 3 x = 0
f
- 8 x^{2} + 3 x = 0
g
- 2 x^{2} + 9 x + 5 = 0
h
3 x^{2} - 8 x + 2 = 9 x - 7
2

Solve the following equations using the quadratic formula:

a

x^{2} + 5 x + 6 = 0

b

x^{2} - 5 x + 6 = 0

c

x^{2} - 5 x + \dfrac{9}{4} = 0

d

x^{2} - 8 x + \dfrac{55}{4} = 0

e
2x^2 + 6x - 8=0
f
4x^2 - 10x +4=0
g

4 x^{2} + 7 x + 3 = 0

h

4 x^{2} - 17 x - 15 = 0

i

- 6 + 7 x + 5 x^{2} = 0

j

- 6 - 13 x + 5 x^{2} = 0

k

- 20 - 11 x + 3 x^{2} = 0

l

- 20 + 21 x + 5 x^{2} = 0

3

Solve the following equations, leaving your answer in surd form:

a

x^{2} - 7 x + 9 = 0

b

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

c

- 2 x^{2} - 15 x - 4 = 0

d
3x^2 + 9x - 4=0
e

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

f

- 5 x^{2} - 15 x + 3 = 0

g

5 x = \left(x - 5\right) \left( 3 x + 3\right)

h

\dfrac{3 x + 1}{3 x - 1} - \dfrac{3 x - 1}{3 x + 1} = 5

4

Consider the equation 2 x^{2} = 14 .

a

Solve for x using the quadratic equation. Give your answer in surd form.

b

Give x as decimal number correct to three decimal places.

5

Solve the following equations, rounding your answers to three decimal places:

a

x^{2} + 3 x - 6 = 0

b

x^{2} + 7 x - 3 = 0

c

4 x^{2} + 7 x + 2 = 0

d

5 x^{2} + 9 x + 2 = 0

e

2 x \left(x - 4\right) = 3 x + 1

f

3 x \left(x + 4\right) = - 3 x + 4

g

7.1 x^{2} + 5.3 x - 1.5 = 0

h

1.8 x^{2} + 5.2 x - 2.3 = 0

6

Solve the following expression for m: 10 - 6 m + 2 m^{2} = m^{2} + 8 m + 9

7

For each of the given solutions to a quadratic equation:

i

Find the values of a, b and c.

ii

Write down the quadratic equation that has these solutions.

a
x = \dfrac{- 5 \pm \sqrt{5^{2} - 4 \times \left( - 7 \right) \times 10}}{2 \times \left( - 7 \right)}
b
x = \dfrac{- 22 \pm \sqrt{484 - 108}}{18}
Applications
8

An object is launched from a height of 80 \text{ ft} with an initial velocity of 107 \text{ ft/s}.

After x seconds, its height, h, is given by:

h = - 16 x^{2} + 107 x + 80

Find the number of seconds, x, after which the object is 30 \text{ ft} above the ground. Round your answer to one decimal place.

9

An object is launched from a height of 8 \text{ m} with an initial velocity of k \text{ ms}^{-1}.

After t seconds, its height, h(t), is given by:

h(t) = -2 t^{2} + k t + 8

The object was in flight for 4 seconds before reaching the ground. What was its initial velocity k ?

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