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Algebra 69 - Quadratic Equations

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.