Quadratic Formula and Discriminant Notes
Quadratic Equations
Standard Form
Vertex Form
Intercept Form
We have solved by graphing and estimating the roots. (Standard form or vertex form)
We have solved by factoring. (Standard form or intercept form)
We have solved by using the square root property. (Vertex form)
Now we will solve by the quadratic formula. (standard form)
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
To solve a standard form quadratic equation by using the quadratic formula
• put it in order and make sure it is set equal to 0 ax2+bx+c=0
• Now make a list of the values
> a=
> b=
> c=
• Next we find the value of the discriminant which is b2 ­ 4ac call this D
• Now the solutions can be found by substituting those four values (a, b, c, D) into the quadratic formula below (Notice there are really two answers one for the + and one for the ­. You should list them separately.) Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Sample problem solved by quadratic formula
First make sure it is in order and = to 0.
Now list a, b, c, and find D
a = 1
D = b2 ­ 4ac
b = 6
= 36 ­ 4(­16)
c = ­16
=100
So the solutions are (remember to solve BOTH + and ­ to get two ansers)
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Now you try this one
a =
D =
b =
c =
x = Check your answers They should be x = ­11 or ­3/2
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Sometimes there is only one root. This happens when the value of D is 0.
Try this one and see how that works.
a = D =
b = c = x =
The answer should only be 8.
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Sometimes the discriminant, D, does not simplify to an integer and your answer has a square root in it. You can leave it as long as it is simplified. Try this one.
a = D =
b = c = x =
The answer should be
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Sometimes the discriminant, D, is not a positive number. This means we have a negative under the radical and our answers must include i. They are imaginary. So there are NO real roots only 2 imaginary roots. We still simplify the radical as much as we can and write the negative as an i. Try this one.
Hint: Is it in order and = to 0 ?
a = b = c = x =
Check your answer
It should be Algebra 2 Academic
D =
Quadratic Formula and Discriminant Notes
Recall the following from a previous class.
Solving Quadratics Graphically
Quadratics may have
NO REAL SOLUTIONS
ONE REAL SOLUTION
TWO REAL SOLUTIONS
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
We can determine the number of solutions without graphing by using the value of the discriminant, D.
There are only 3 possible values of D
It can be zero, positive or negative.
D = 0
D < 0
means means one real solution
no real solution
and it is rational.
the roots are imaginary conjugates
D > 0
means two real solutions
If D is a perfect square they are rational. If not they are irrational.
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Read the following examples and then try the problems at the bottom.
Your answers should be
5A) D = 44 5B) D = 289
so 2 real roots so 2 real roots
because 44 is not because 289 is
a perfect square they a perfect square
are irrational 172 = 289 they
are rational
Algebra 2 Academic
Quadratic Formula and Discriminant Notes
Begin the worksheet of practice problems.
It must be handed in when you are done.
You can finish for homework, if it is not completed by the end of class.
Algebra 2 Academic