We're going to continue our discussion of solving quadratic equations by using the quadratic formula
but this time we're really going to hone in and
Identifying the number in type of solutions for each of our quadratic equations and
this is called using the discriminant now the discriminant is
The involved piece of our quadratic formula which is b squared minus 4ac. So that is
the discriminant
Now the discriminant tells us if we're going to have real solutions or non real solutions
So remember the discriminant b squared minus 4ac is the part. That is underneath
our
square root
so if
The discriminant is less than zero. That means it's negative
The square root of a negative is going to result in non real
or
imaginary
Solutions. So for example
With two times x squared minus 3x plus five set equal to zero
B squared minus 4ac. Well B is
negative three
so negative three squared is none and
then
minus four
times a a is two
And C is positive five. So we're going to be subtracting
four times 2
times 5 which is 40 and
9 minus 40 produces a negative number which is why our solutions will be
imaginary the square root of a negative is
imaginary
Now if our discriminant is equal to zero we're going to have one real solution
But remember every quadratic equation has two solutions
so this means we're actually going to have
One solution that happens twice. This is also known as a double root
So 4x squared plus 6x plus 9 set equal to 0
B is 6 and 6 squared is 36
a
is 1
C is 9 and 4 times 1 times 9 is also
36 which produces an outcome of 0 for the discriminant and remember the square root of 0 is
0 and
Negative 6 plus or minus 0 gives us negative 6 and negative
6 divided by 2 is negative 3
so we have
one solution of x equals negative 3 which actually happens twice and I want us to see why that is so
If we think about x squared plus 6x plus 9
There's actually factors in the factors of x squared plus 6x plus 9 or X plus 3
times X plus 3
so, can you see that we'll have a solution of x equals negative 3 and it happens twice and
Then our last one I like to get a little bit more specific than this if
Our discriminant is positive
That means our solutions will be real but remember they could be real and rational
or or they could be real and
Irrational and we'll talk about the difference between those two
notice for this example, our discriminant is 57 and
57 is positive. So it says our solutions will be real
But 57 is not a perfect square. So it says our solutions will be irrational
You can't do anything with the square root of 57
So for each of the examples we're going to look at we are not going to be solving these quadratic equations
We're simply trying to figure out the number and type of solutions. We're going to have and that is by identifying
the discriminant
so the discriminant is
found by identifying a
Which in this case three?
B
Which is negative five and C which is four in
determining what b squared minus 4ac
Actually is
So we have negative five notice. I'm putting that negative five in parenthesis because I'm squaring that value
minus four times a times C
And negative five squared is positive 25
minus four times three is 12 and 12 times 4 is 48 and
25 has a value that is less than 48 therefore. We have a negative
Which means
We're going to have two solutions
But these are going to be both
imaginary remember the discriminant is the piece underneath our square root and
A square root of a negative is not real. We have two imaginary solutions
For our next example
We need to identify the number and type of solutions by identifying a
B and C
and
determining the discriminant
The discriminant is b squared minus 4ac
So we have B, which is negative 7 in parenthesis squared
Minus 4
times 6
Times 2 now because a and C
are both positive we again have the possibility of
having
imaginary numbers because we're subtracting a positive
We don't know if this is going to result in a positive number or a negative
It all depends on what B squared is and how it relates with this for AC
So negative 7 squared is 49
Minus
4 times 6 is 24 and
24 times 2 is 48
Which results in a positive 1 so this tells us because this is positive
That our solutions will be real now. I like to take this a step further though
note
Positive 1 it's not just positive. It is also a
perfect square because 1 times 1 1 squared is 1
so that means our solutions will not only be real but they will be
rational
because the square root of 1 can simplify to be 1
Our next example
We have the negative of x squared minus 5x plus 4, so we're going to know
That a is equal to negative 1
B is
Equal to negative 5 C is equal to positive 4
So B squared minus 4ac
Gives us negative 5 squared minus
4 times negative 1
Times positive 4 which means right here right now. I know that we are going to have
Two
real solutions
Now, how do I know my solutions are going to be real?
the reason I know this is a negative squared is a positive and I have minus a
negative and
Subtraction of a negative becomes positive. So this is plus 16
Which results in 41 and 41 is a positive number so that reconfirms
That my solutions will be real but because 41 is not a perfect square that means our solutions
will be
irrational
Our next example
Number and type of solutions here. Where a
is equal to 1
B is equal to negative 4 and C is equal to positive 4
So for our discriminant of b squared minus 4ac
We have negative 4
squared minus
4 times 1 times 4 and
Negative 4 squared is 16
Minus 4 times 1 times 4 is also 16
Which is 0 which means we will have one
real
Solution and this is again what we call a double root
one solution that happens twice
For problem number five
We have quite a bit of work we need to do in order to be able to
Determine the number and type of solutions we have so we need to clean this up and we need to get it in general form
so that we can identify a
B
And C so we need to distribute and combine like terms so distributing we have three times x squared
minus 6x
equals 5x plus four, but remember we need to set this equal to zero and
So we want to leave our 3x squared alone. It's in good position
minus 6x
We need to combine this with this 5x and get them both on the same side, so we're going to subtract the 5x and
We need to set this equal to zero. So we need to also subtract the four
So we have three times x squared
minus 11x minus
four
and now we can identify a
B
And C so
That we can determine the discriminant which is b squared minus 4ac
So b is negative 11 and we're going to square that
minus 4
times a times
C
So negative 11 squared is 121 and is this going to be plus or minus?
So we have 4 times 3 times negative 4, which is negative 48
But we're subtracting that so that becomes plus 48
and
121 plus 48 is
169
So because this is positive this is going to tell us we're going to have two real solutions
But I like you to take it a step further. Is this a perfect square? Yes. No
if you were to
Put this in your calculator
The square root of 169 is that going to give you an integer
Or is it a non-repeating nonterminating
Decimal
So 169 is a perfect square, isn't it? It's 13 squared
And so that means we're going to have two real rational solutions
Now you have several questions where you're asked to determine the number and
type of solution so for each of these you are going to identify a
B
and C in
order to determine the discriminant and
Then you're going to have one of four categories
Either you're going to have two
real rational solutions
Or you're going to have two real irrational solutions
Or you're going to have one real solution
Or you're going to have two imaginary
Solutions
So you're going to do that for each of these questions
the first question
Determine the number and type of solutions for four times. Pardon me 4 times x squared
plus 3x minus 14 equals 0
For question two number and type of solutions for six times x squared
Plus 13x equals negative two
And for the third question number and type of solutions for 3x
times X minus 2
equals 2x minus
8
