Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture, we introduced a general method of solving quadratic equations
called "completing the square".
Before the method of completing the square was developed
only very limited types of quadratic equations could be solved.
This method eliminated those limitations
allowing the solutions of any quadratic equation to be found.
As mathematics progressed, this methodology was eventually reduced to a formula.
Given any general-form quadratic equation "a x-squared plus bx plus c equals zero"
the values for the constants a, b, and c are entered into the formula
that then produces the solutions to the equation.
In other words, given the values of a, b, and c
this formula tells us the values of x
that will cause the quadratic expression to have a value of zero.
In this lecture, we will show how this "quadratic formula" can be derived
from the process of completing the square
and show why the quadratic formula is so useful.
To derive the quadratic formula, we start with a quadratic equation in general form.
The first step is to eliminate the coefficient of x-squared
by dividing both sides by that coefficient.
Next, we change the quadratic expression into a perfect square
by changing its constant term as we saw in the lecture "Completing the Square".
In that lecture, we saw that if the value of a quadratic expression's constant term
is the square of one-half its x coefficient
then that quadratic expression will be a perfect square.
As we also saw, the quadratic's constant term can be replaced with this value
by first subtracting the constant term from both sides of the equation
and then adding the new constant value to both sides.
Since this makes the quadratic expression on the left a perfect square
it can be written as x
plus one-half the quadratic's x-coefficient
quantity squared.
Then on the right side, instead of squaring the entire fraction "b over 2a"
we can square all the terms in the fraction separately.
The two fractions on the right can then be combined
by creating a common denominator
multiplying the top and bottom of the fraction "c over a"
by 4a.
The denominator "4a times a" is "four a-squared".
And since the two fractions now have a common denominator
a single fraction can be formed by combining their numerators.
Now, to eliminate the square on the left side of the equation
we take the square root of both sides of the equation.
As we have seen, since the square root of the square of an expression
is the same as the absolute value of that expression
the square root of the square can be replaced with the absolute value operation.
The absolute value of the left side of the equation can then be eliminated
if we add a plus-or-minus sign in front of the right side.
Then on the right side, instead of taking the square root of the entire fraction
we can take the square root of its numerator and denominator separately.
And taking the square root of the denominator, four a-squared
we get positive or negative 2a.
However, since the entire fraction can already be either positive or negative
the plus-or-minus sign in the denominator is redundant and can therefore be eliminated.
We then move "b over 2a" to the right side by subtracting it from both sides.
And since both fractions on the right have a common denominator
a single fraction can be formed by combining their numerators.
We now a formula that gives the values of x
that are the solutions to any general form quadratic equation.
This famous formula is known as "the quadratic formula".
Although the quadratic formula solves quadratic equations
using the same logic as "completing the square"
it is more straightforward to use.
As an example, in the lecture "Completing the Square Part One"
we solved the quadratic equation "x-squared plus 9x plus 12 equals zero".
Using the technique of completing the square, we arrived at the solutions
negative nine plus-or-minus the square root of 33 over two.
Let's now solve this same quadratic equation using the quadratic formula.
This quadratic equation is in general form
and the constants a, b, and c
are "one"
nine
and twelve.
Substituting these values into the quadratic formula
we replace "a" with "one"
b with nine
and c with twelve.
So the solutions to the quadratic equation are quickly produced.
As we saw in "Completing the Square Part One" the graph of this quadratic function
crosses the x-axis at the two values of x that are the solutions to the quadratic equation.
We have seen that quadratic equations may have two real solutions
a single real solution
or no real solutions.
So let's see what happens when we use the quadratic formula
to solve a quadratic equation with a single solution.
The quadratic function "four x-squared plus 20x plus 25"
is a perfect square.
And like all perfect square quadratic functions
it has a single x-intercept and thus a single zero.
Now let's form a quadratic equation by equating this function to zero.
Then, using the quadratic formula
we can find the solutions to this equation and thus the zeros of the quadratic function.
This quadratic equation is in general form and its constants a, b, and c
are 4
20
and 25.
We now substitute these values into the quadratic formula and simplify.
Replacing "a" with four
b with 20
and c with 25
we see that the numbers under the square root sign reduce to zero.
And so the plus-or-minus sign adds or subtracts a value of zero
resulting in a single x value instead of two x values.
The quadratic formula therefore produces a single solution "negative five-halves"
that corresponds to the single zero of the quadratic function
"four x-squared plus 20x plus 25".
So far, we have seen that the quadratic formula produces two solutions to a quadratic equation
when the corresponding quadratic function has two x-intercepts
and a single solution when the function has a single x-intercept.
But what happens when the function has no x-intercepts?
To find out, let's apply the quadratic formula to the function "x-squared plus 6x plus 10".
Once again, we equate this function to zero to form a quadratic equation
and substitute the values for a, b, and c into the quadratic formula.
Replacing "a" with "one"
b with six
and c with ten
we see that the numbers under the radical sign reduce to negative four.
And since negative numbers have no real square roots
this equation has no real solutions.
As we have just seen
the value of the expression under the square root sign in the quadratic formula
tells us important information about the quadratic equation we wish to solve.
This expression, called the "discriminant"
tells us the number of real solutions that exist for the quadratic equation.
If the discriminant is positive
the quadratic equation has two distinct real solutions.
If the discriminant is zero
the quadratic equation has one real solution.
And if the discriminant is negative
the quadratic equation has no real solutions.
In our first example
with the quadratic equation "x-squared plus 9x plus 12 equals zero"
the discriminant
was "9 squared"
minus "4 times 1 times 12"
or 33.
Since the discriminant is positive
this quadratic equation has two distinct real solutions
and the corresponding quadratic function has two distinct real zeros.
Likewise, in our second example
with the quadratic equation "four x-squared plus 20x plus 25 equals zero"
the discriminant was "20 squared"
minus "4 times 4 times 25"
which reduces to zero.
Therefore, this quadratic equation has one real solution
and the corresponding quadratic function has a single zero.
And in our third example
with the quadratic equation "x-squared plus 6x plus 10 equals zero"
the discriminant was "6 squared"
minus "4 times 1 times 10"
or negative four.
Since the discriminant is negative
this quadratic equation has no real solutions
and the corresponding quadratic function has no real zeros.
As we have seen, this is because no real number exists
that is the square root of a negative number.
However, in the next lecture, we will introduce an extension to the real number system
that allows the square root of negative numbers to be represented.
This extension is created by a mysterious new set of numbers called "imaginary numbers".