Hello. I'm Professor Von Schmohawk
and welcome to Why U.
So far we have introduced quadratic functions
and have seen that the graph of a quadratic
function in a single variable is a parabola.
If we set a quadratic function equal to zero
we produce a "quadratic equation".
Although quadratic functions and
quadratic equations look similar
there are some important differences.
For example, let's take the quadratic function
"x-squared - 4".
Like all quadratic functions
the graph of this function is a parabola.
Now if we set this function equal to zero
we produce a quadratic equation.
This equation has two solutions
"x equals 2"
and "x equals negative two".
So although the graph of the function
"x-squared - 4" is a parabola
the solutions to the quadratic equation
are the two points where the parabola
intersects the x-axis
also known as its x-intercepts.
These points are called
the "zeros" of the function.
In general, the zeros of a function
are the input values
that cause the function's output
to have a value of zero.
So for a function of x, the values of x which
cause the function to produce a value of zero
are the zeros of the function.
These zeros are sometimes called
the "roots" of the function.
Finding the zeros of a linear function is easy
since unless the function's graph
is a horizontal line
it will always have a single x-intercept
and thus a single zero.
To find the zero of a linear function
we determine its x-intercept
by setting the function equal to zero
and then solving the equation for x.
On the other hand, a quadratic function of
a real variable may have two x-intercepts
one x-intercept
or no x-intercepts.
But solving a quadratic equation can be much
more complicated than solving a linear equation.
The problem of finding the solution
to simple quadratic equations
was encountered at least 4000 years ago
by both the Babylonians
and the Egyptians.
Since ancient peoples did not have the benefit
of algebraic equations
they described their procedures
using words and pictures.
In those ancient times, problems involving
squares of quantities typically came up
when calculating the areas and dimensions
of structures or plots of land.
For example, in Babylonia, a standard
measurement of length was the "nindan".
And for a standard unit of area,
one square nindan was defined as one "sar"
equal to about 36 square meters.
A problem involving quadratics
that the Babylonians might have to solve
would be to find the length of the sides
of a square storage chamber
which must have an area of 16 "sar".
In modern algebraic terms,
we could represent this problem
by the equation "x-squared equals 16"
where x represents the unknown length
of the chamber's side.
This is equivalent to solving the quadratic
equation "x-squared - 16" equals zero.
In this case, since length must
always be a positive value
the value of x can be found
by taking the square root of 16
which the Babylonians and Egyptians did
by referring to tables of square roots
written on clay tablets or papyrus scrolls.
A more difficult problem might be
to find the length of the sides of a
square storage chamber
with an additional 3-nindan long extension
where the total storage area must be 28 sar.
This problem could be represented
by the equation "x-squared"
"plus 3x"
"equals 28".
Or subtracting 28 from both sides
the equation could be written as
"x-squared + 3x - 28" equals zero.
Solving this quadratic equation
is more complicated than just taking a square root
so procedures had to be developed
by the Babylonians
to solve more general types
of quadratic problems like this one.
In more recent times, many types of problems
have been discovered
which can be represented by quadratic equations.
For instance, the trajectory of a free-falling
object can be described by a quadratic equation.
As an example, if a cannonball is fired
from a cannon 12 meters above the ground
and the cannonball has an initial
vertical velocity of 80 meters per second
then according to the laws of Newtonian mechanics,
the height of the cannonball above the ground
would be given by the function
"negative 5 t-squared + 80t + 12"
where the constants in this quadratic expression
are determined by the physical parameters
of the example
and t is the time elapsed after the cannon is fired.
If we wish to find the time it takes for the
cannonball to hit the ground
we set this expression for the vertical height
equal to zero
and solve the resulting quadratic equation for t.
Solving this equation requires a mathematical
technique more powerful
than anything that the Babylonians or Egyptians
had in their mathematical bag of tricks.
In the next several lectures, we will explore
techniques for solving quadratic equations
and see how these techniques evolved
through the centuries
eventually allowing us to find the solutions
to any quadratic equation.
