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When we have a quadratic, we normally specify it in the form <math>ax^2+bx+c=0</math> | When we have a quadratic, we normally specify it in the form <math>ax^2+bx+c=0</math> |
Revision as of 18:15, 1 June 2011
One root equations
Written by Justin Stevens
Contents
Deriving the quadratic formula
When we have a quadratic, we normally specify it in the form
The way to find a formula for the roots of the equations goes as follows:
\[\sqrt{(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\] (Error compiling LaTeX. Unknown error_msg)
We have two real roots when
We have complex roots when
We have one root when
We are going to analyze the case when
is called the discriminant.
Square both sides to get:
Add
to both sides to get:
Let’s evaluate cases now.
1.2 The discriminant when a= c
First we evaluate the case
When a=c, we have:
Take the square root of both sides to get:
Therefore when and
we are going to have one root for the equation.
Example 1:
Example 2:
When and
we can write the problem as follows:
Using the quadratic formula, we get:
A simpler way to do this, is to start off by dividing by a. We get:
If we want to write this as one equality, we can do as follows:
Divide by 2 on the second equation to get:
Therefore we have
So, when we will have one root, and that one root will be –1.
1.3 The discriminant when a=b
When we get:
The roots are going to be
Therefore we have:
or
However, if (a,b)=0, we would be left with
This is only a constant term left.
So, our only case is
Example 1:
Example 2:
Let’s find a general formula for this.
If , we have:
Using the quadratic formula, we get:
Another way to do this, is to start off by dividing by c.
Divide by 4 to get:
Therefore when we will have one root which will be
1.4 The discriminant when b=c
, therefore
The roots are going to be which gives us:
Since we get:
Divide by a to get:
However, since (a,b,x)=(0,0,0) this is just a case with only a quadratic term, and shouldn’t be considered.
Let’s try the second case of which is going to be
We have:
Example 1:
Example 2:
Let’s try an in-general example of when
Using the quadratic formula, we get
We can also start off by dividing by a to get:
Therefore when in
, the root is going to be
1.5 Conclusion
When we will have one root. That one root will be –1.
When
we will have one root. That one root will be
When
we will have one root. That one root will be –2
Contributors
Thanks to our contributors: AIME15 El_Ectric BOGTRO bzprules El_Ectric SuperNerd123