Okay here we're talking about the quadratic formula.
Which everyone should have memorized by now, I would hope.
It's... if you have a quadratic equation
ax squared plus bx plus c equals zero, we call that standard form of the equation.
Then your solutions, you get two of them, are x is
opposite of b plus or minus square root of b squared minus 4ac all over 2a.
At the end of this video I'll include a link to the song.
I didn't do the song, but you can go watch the song and you can sing it along if you want.
Keep in mind if you're gonna use quadratic formula,
you have to be in standard form. You have to have your equation...
Simplify it over here on the left side equal to zero.
Okay. This this piece here underneath the radical: b squared minus 4ac.
That tells us what kind of x-intercepts
our function f of x equals ax squared plus
bx plus c would have.
Remember, that's what we're really doing is finding the x coordinate of x intercepts when we're using quadratic formula.
So this piece under the radical tells us whether we have two x-intercepts, one, or none.
If this piece under the radical is positive,
then when we take the square root we get a real number.
If this piece under the radical is zero, square root of zero is zero, and we get a plus or minus
zero, we get what's called a repeated root.
And if this piece is negative,
you can't take a square root of a negative and get a real number.
We get complex or imaginary numbers, and we don't get x-intercepts.
Okay, that's all laid out here.
The discriminant
discriminates between the different types of quadratics we have.
It's b squared minus 4ac, and if it's positive you get two real,
unique
solutions to the equation
and the graph has two x-intercepts.
If the discriminant is zero you only get one,
solution, it's a repeated solution.
And you only get one x-intercept.
And if the discriminant is negative you get complex solutions to your equation
and the graph has no x-intercepts.
Okay, so in this particular case we would have a positive
discriminant, we would have two solutions
to the equation ax squared plus, bx plus c equals zero.
And this function f of x would have two x-intercepts right here.
(Negative 3, 0) and (2,0) and they're unique, they're different.
Here's a case where the discriminant is equal to 0.
That's when that the parabola sits right on the vertex.
The vertex is right on
the x-axis.
So the equation ax squared plus, bx plus c equals zero would only have one solution.
This one in particular would have x equals 2, and x equals 2, the same solution twice.
Call that a repeated solution.
The graph only has one x-intercept, which happens to be the vertex, which it always will be.
(2,0)
And in this case we have a negative discriminant.
That's when we would be taking square roots of negative numbers.
Those do not correspond to x-intercepts, this graph has no x-intercepts at all.
The equation ax squared plus, bx plus c equals zero still has solutions.
But they're not real, they're complex solutions,
and you can't graph complex numbers here in the real plane.
Okay an example of using
the quadratic formula to get our solutions.
Say we want to solve 2x squared plus 4x minus 5 equals 0
First we identify a, b, and c.
a is 2, b is 4, c is negative 5.
So we want x equals the opposite of b plus or minus the square root of
b squared minus 4 times a times c
all over
Twice a.
And then it's just arithmetic.
I remember order of operations, 4 squared is 16.
Negative 4 times a 2 times a negative 5 is a plus 40 and
then 16 plus 40 is 56.
Here are the solutions to this equation.
x is negative 4 plus or minus root 56 over 4
That can't be simplified.
That graph would look like this.
I notice this graph points up, we know that
by looking at a.
a tells us if the graph points up or down.
a is positive 2 so the graph points up.
Those x-intercepts, we had two of them, are here.
Those x values are
these two different x values here
of course the y-values are 0.
x-intercepts always have y-values of 0.
Ok here's another example
We're solving x squared plus 10x plus 25 equals 0.
a is 1, b is 10, c is 25.
Remember a, b, and c are just numbers. They don't have x's in them
You'll never have any x's written over here inside quadratic formula.
So we plug everything in and we get, under the radical, 100 minus 100 which is 0.
So we get negative 10 plus 0.
That's negative 10.
Negative 10 minus 0.
That's also negative 10.
Over 2.
And of course negative 10 over 2 is negative 5.
We get the same
answer twice.
That means on the graph
x
equals negative 5
is the x coordinate of my x intercept.
The x intercept is (negative 5 comma 0)
The solutions to this equation,
there are two of them,
quadratics always have two solutions.
They just both happen to be the same number.
They're both x equals negative 5.
Ok that's a case where the discriminant was equal to zero.
Here's another case where the discriminant is negative. Now, we'll get a negative under the radical.
So here a is 1, b is 4, and c is 6.
[That's a typo right there that should not be a negative]
[So ignore that negative.]
We get x equals
opposite of b plus or minus square root of b squared,
this should just be 4 squared
minus 4 times a times c all over
2a.
Under the radical we get 16 minus 24, which is negative 8.
Square root of a negative gives us an i outside.
Square root of 8, you might remember from simplifying radicals
would be 2 times square root of 2.
So our final answers are
x equals negative 4 plus or minus 2i root 2 over 2.
Two solutions, quadratics always have two solutions.
These two just happen to be complex
so the graph will have no x-intercepts.
Ok that is quadratic formula. Click on any of these videos to watch related topics.
