1.
Determine whether solutions exist for each of the following quadratic
equations. Where they do find the solution(s).
Firstly determine whether solutions exist using the following criteria:
b 2 − 4ac > 0
Two solutions
2
b − 4ac = 0
One solution
2
b − 4ac < 0
No solution
Secondly find the solution where possible using the formula:
x=
(i)
− b ± b 2 − 4ac
2a
x2 − 2x = 0
a=1, b=-2, c=0
b 2 − 4ac = (− 2 )2 − 4(1)(0) = 4 > 0
two solutions exist
− b ± b 2 − 4ac 2 ± 4 2 ± 2
=
=
2a
2(1)
2
2+2
x=
=2
2
2−2
x=
=0
2
x=
(ii) (3x − 6 )( x + 1) = 0
Multiply out the quadratic
3x 2 − 3x − 6 = 0
Divide across by 3
x2 − x − 2 = 0
a=1, b=-1, c=-2
2
b 2 − 4ac = (− 1) − 4(1)(− 2 ) = 9 > 0
two solutions exist
− b ± b 2 − 4ac 1 ± 9 1 ± 3
=
=
2a
2(1)
2
1+ 3
x=
=2
2
1− 3
x=
= −1
2
x=
(iii) 9 x 2 − 24 x + 16 = 0
a=9, b=-24, c=16
b 2 − 4ac = (− 24 )2 − 4(9 )(16) = 576 − 576 = 0
x=
− b ± b 2 − 4ac 24 ± 0 24
=
=
= 1.33
2a
2(9 )
18
one solution
(iv) 3x 2 + 2 x + 3 = 0
a=3, b=2, c=3
b 2 − 4ac = (2 )2 − 4(3)(3) = 4 − 36 = −32 < 0
no solution
(v) 2 x 2 + 11x − 21 = 0
a=2, b=11, c=-21
2
b 2 − 4ac = (11) − 4(2 )(− 21) = 121 + 168 = 289 > 0
two solutions
− b ± b − 4ac − 11 ± 289 − 11 ± 17
=
=
2a
2(2 )
4
− 11 − 17
− 11 + 17
x=
= 1.5
x=
= −7
4
4
x=
2
(vi) − 2 x 2 + x + 10 = 0
a=-2, b=1, c=10
b 2 − 4ac = (1)2 − 4(− 2 )(10) = 81 > 0
two solutions
− b ± b 2 − 4ac − 1 ± 81 − 1 ± 9
=
=
2a
2(− 2 )
−4
−1− 9
−1+ 9
x=
= −2
x=
= 2.5
−4
−4
x=
2 A firms demand function for a good is given by P = 107-2Q and their total
cost function is given by TC = 200+3Q .
i) Obtain an expression for total revenue profit in terms of Q
Total Revenue = P.Q
TR = (107-2Q)*Q = 107Q-2Q2
Profit = TR-TC
Profit = 107Q-2Q2-200-3Q = -2Q2+104Q-200
ii)
For what values of Q does the firm break even
Firm breaks even where Profit = 0
-2Q2+104Q-200 = 0
a = -2, b=104, c=-200
Q=
− 104 ±
Q = 2, Q = 50
(104 )2 − 4(− 2)(− 200) − 104 ±
=
2(− 2 )
10816 − 1600 − 104 ± 96
=
−4
−4
iii)
Illustrate the answer to (ii) using sketches of the total cost function,
the total revenue function and the profit function
2000
TC / TR / Profit
1500
Proft =
1150
1000
500
TC
TR
0
0
10
20
30
40
Q = 26
50
Profit
Profit
-500
Note: Break even where Profit = 0 or TR=TC.
iv)
From the graph estimate the maximum profit and the level of output
for which profit is maximised
Maximum profit at max point on profit curve.
Max profit = 1150 at Q = 26
3. What is the profit maximising level of output for a firm with the marginal
cost function MC = 1.6Q2-15Q+60 and a marginal revenue function MR =
280-20Q?
Profit is maximised where MR=MC
280-20Q = 1.6Q2-15Q+60
1.6Q2+5Q-220=0
a=1.6, b=5, c=-220
Q=
−5±
(5)2 − 4(1.6)(− 220) − 5 ±
=
2(1.6)
25 + 1408 − 5 ± 37.85
=
3.2
3.2
Q = 10.27, Q = −13.39
Profit maximising level of output is Q = 10.27 (can’t have negative output)
Q
60
4. The demand function for a good is given as Q = 130-10P. Fixed costs
associated with producing that good are €60 and each unit produced costs an
extra €4.
i) Obtain an expression for total revenue and total costs in terms of Q
TR = P.Q
Q = 130-10P
10P = 130-Q
P = 13-Q/10
TR = (13-Q/10)*Q = 13Q-0.1Q2
TC = FC+VC
TC = 60+4Q
ii)
For what values of Q does the firm break even
Firm breaks even where TR = TC
13Q-0.1Q2=60+4Q
-0.1Q2+9Q-60=0
a=-0.1, b=9, c=-60
Q=
−9±
(9)2 − 4(− 0.1)(− 60) − 9 ± 81 − 24
=
− 0.2
2(− 0.1)
=
− 9 ± 7.55
− 0.2
Q = 7.25, Q = 82.75
iii)
Obtain an expression for profit in terms of Q and sketch its graph
Use the graph to confirm your answer to (ii) and to estimate maximum
profit and the level of output for which profit is maximised
Profit = TR-TC
Profit = 13Q-0.1Q2-60-4Q=-0.1Q2+9Q-60
iv)
200
Profit
150
Profit Max
Profit = 143
100
Profit
50
Q
0
0
-50
-100
10
Break Even
Q = 7.25
20
30
40
50
Profit Max
Q = 45
60
70
80
Break Even
Q = 82.75
90