2.3 Solving Complex Equations
This section will involve solving equations, but the algebraic manipulations will be a little more
complex. It would be beneficial at this point to review radicals (inverse operations) and how to
accomplish this on the calculator.
SQUARING A NUMBER comes from the concept of a SQUARE. By definition, a square is
2-dimensional and has the same measure on each side.
2x2 = 22
4 square units
3x3 = 32
9 square units
4x4 = 42
16 square units
AND SO ON!
Numbers like 4, 9, 16, 25, etc. are called “perfect square” because they represent a square
area.
Thus: 92 is verbalized as “nine squared.”
The reverse (or inverse) of squaring a number is taking its “square root” using the symbol
When we use this symbol, we are given the area of a square and looking for
the measure of one of its sides.
EXAMPLE:
Area of the square.
x2 undoes x2
49  7
Length of one side of the square.
They are REVERSE operations.
CUBING A NUMBER comes from the concept of a CUBE. By definition, a cube is 3dimensional and has the same measure on each edge.
2x2x2 = 23
8 cubic units
3x3x3 = 33
27 cubic units
4x4x4 = 43
64 cubic units
AND SO ON!
Numbers like 8, 27, 64, 125, etc. are called “perfect cubes” because they represent a the
volume of a 3-dimensial rectangular solid that measures the same on each side.
Thus: 93 is verbalized as “nine cubed.”
The reverse (or inverse) of cubing a number is taking its “cube root” using the symbol
When we use this symbol, we are given the volume of a cube and looking
for the measure of one of its edges.
EXAMPLE:
3
729  9
Volume of the cube.
3
x3 undoes x3
Length of one edge of the cube.
They are REVERSE operations.
This same pattern exists for all exponents.
Square
Root x
Square x 2
x
multiply a
number
the inverse
times itself 2 of squaring a
times
number,
2
what number
x means
times itself 2
xx
times equals
x
Cube Root
3
x
To the
power
of 4 x 4
the inverse
of cubing a
number,
what number
times itself 3
times equals
x
multiply a
number
times itself
4 times
4
x means
xxxx
Cube x 3
multiply a
number
times itself
3 times
x 3 means
xxx
1
12  1
1 =1
13  1
3
1 1
14  1
2
22  4
4 2
23  8
3
82
2 4  16
4th root
4
x
the inverse
of raising a
number to
the 4th
power, what
number
times itself 4
times equals
x
4
4
1 1
16  4
The following values can
be referred to as “perfect
squares.”
Perfect Cubes
Perfect 4ths????
1
1
1
1
2
4
8
16
3
9
27
81
4
16
64
256
5
25
125
625
And so on……..
You can manually find the root of a number and, if you have your heart set on it, there is a
relatively boring utube video (I’m sure there is more than 1). So, google your heart out. I
watched part of one recently and that’s enough to satisfy my thirst for quite some time.
It is much easier to use your calculator to:
Raise a value to a power
Practice:
10 2 
Or
Find a “root” (round to 1 decimal if necessary)
100 
103 
3
1000 
10 4 
4
10000 
Example 1:
The formula for the Radius of an arch window can
be used in another interesting application.
The formula to calculate the radius (R) of a portion of a circle is
W 2  4H 2
R
8H
W = width of the water
H = height of the water
Find the width of the water in a 12 inch radius pipe if it is 8 inches at its deepest point.
Example 2:
What is involved in determining the size of a car’s engine?
An engine’s volume or Displacement (D) is D 
 b 2sc
4
D = engine displacement measured in cubic centimeters
b = bore (diameter of the cylinder) measured in centimeters
s = stroke (distance that the piston travels) measured in centimeters
c = number of cylinders.
Find the bore necessary for a 6-cylinder engine with a 6-in stroke with 278 cubic inches of
displacement, rounded to one decimal place.
Example 3:
bd 3
The moment of inertia (I) of a beam is I 
.
12
Note: Moment of inertia is a measure of a
beam’s effectiveness at resisting bending based on its crosssectional shape.
I = moment of inertia of the beam measured in inches4
b = width of the beam measured in inches
d = height of the beam measured in inches
Find the height of a beam rounded to the nearest 8th of an inch if
1
b  5 in and I  3250in 4 .
4
Example 4:
Fill in the table of values accurate to three decimal places for the electrical circuit wired in
parallel, using the two primary electrical formulas:
Ohm's Law V = R • I and Watt's Power Formula P = V • I
V = voltage (volts), I = current (amps), R = resistance (ohms), P = power (watts)
What you need to know about parallel circuits:
a. Electricity passes through one or the other resistor.
b. Ω is the symbol for ohm, which is the unit of measurement for resistance R.
c. The subscripts for the letters serve only to
distinguish to which resistor they belong: R1 is
resistor one.
R 1R 2
d.
R1  R 2 = Rtotal
e. V1 = V2 = Vtotal
f. I1 + I2 = Itotal
g. P1 + P2 = Ptotal
Example 5:
What determines how much a beam will flex and bend when it is used
in a house or a bridge?
Moment of inertia is a measure of a beam’s effectiveness at
resisting bending based on its cross-sectional shape.
Note: Deflection is simply a measurement of the amount of bend in a beam.
PL3
The point load deflection (D) of a beam is D 
.
48EI
D = deflection measured in inches
P = weight on the beam measured in pounds
L = length of the beam measured in inches
E = elasticity of the beam measured in pounds per square inch (PSI)
I = moment of inertia of the beam measured in inches4
Find the moment of inertia for a beam rounded to the nearest whole number:
D = 1 in
P = 3250 lbs.
L = 164 in
E = 1,800,000 psi
bd 3
The moment of inertia (I) of a beam is I 
.
12
Note: Design a beam with dimensions that will have a moment of
inertia sufficient to maintain the 1 inch deflection and 3250 pound
load in the initial problem.
Homework: Problems 1-14
Section 2.3:
1.
2.
3.
4.
5.
6.
7.
3.5 amps
9.1 ft
61 parts
24 slats
3.432 in
3 in
127 MPH
8. 2
1”
4
9. 185 in
10. 4.7 kΩ
11. 4,033 lbs
12.
V
I
R
P
Total
12
11.54
1.04
138.48
R1
5.54
11.54
.48
63.93
R2
6.46
11.54
.56
74.55
Total
24
7.084
3.388
170.016
R1
24
3.75
6.4
90
R2
24
3.333
7.2
79.992
Total
9
11.39
.79
102.51
R1
5.92
11.39
.52
67.43
R2
3.08
4.53
.68
13.95
13.
V
I
R
P
14.
V
I
R
P
R3
3.08
7
.44
21.56
*** if you round to 2 decimal places as you go