Transcript 101 Uses of a Quadratic Equation

101 uses of a quadratic equation
Chris Budd
Is the quadratic equation evil?
And does anyone care?
Babylonians
Counted in Base 60
x
Area
ax
2
x
Area
x
x
ax  bx
2
Quadratic equation
ax  bx  c  0
2
Solutions
 b  b  4ac
x
2a
2
x 2
2
X = 1.4142135623730950488…
A4 Paper
Conic sections
Appolonius
Cone
Ellipse
Parabola
Hyperbola
Ellipse
x2
y2
 2 1
2
p
q
Parabola
Hyperbola
y 2  4 px
x2
y2
 2 1
2
p
q
All these curves obey quadratic equations
How conic sections caused a revolution
Copernicus
Tycho Brahe
Kepler
Galileo
Studied how things move and found that they obeyed
quadratic equations
Cars do …..
THUD!!
..
Was the car speeding?
Forensic evidence:
collision damage,
witness statements,
skid marks
s
u
F
distance of skid
speed
brake force
Quadratic equation links
them together
Solving the equation
gives the speed
2
u
s
2F
u  2Fs
So do rugby balls ….
y  bx  ax
2
Parabola
Sir Isaac Newton
Explained the link and also invented the reflecting
telescope based on the parabola
Newton discovered differential equations
2
d u
du
a 2 b
 cu  0
dt
dt
Used to explain almost everything
• Pendulums
• Car suspensions
• Musical instruments
• Radios and TV
Euler finds
the solution
ue
xt
ax  bx  c  0
2
Solve the quadratic equation to solve the differential
equation
And millions of important problems
Quadratic equations take flight
u 2
2
 P  gh
Speed
Pressure
Bernouilli
Chaotic quadratic equations
Bee population x evolves every year
xnew  a xold (1  xold )
population
a=2
Population settles down
population
a=3
Boom and bust
population
a=4
Chaos
Are you still frightened of me?