quadratic formula
Elementary Functions
We have two types of elementary functions, so far:
- Linear functions
- Quadratic functions
One of our goals for analysis is to identify zeros of functions.
Unless the linear function is actually a constant function (which is a linear function), the linear function has exactly one zero. We can identify this unique zero by setting the formula equal to and solving. This is accomplished by combining like terms and isolating the variable on one side of the equation.
We have seen one approach to identifying zeros of quadratic functions.
- Completing the Square
We have a second approach.
The Quadratic Formula
After stepping through the procedure of completing the square, we could have extended the procedure and solved for .
Completing the square gave us
Now, pretend this was equal to in a quadratic equation.
Solve for .
Either
or
- If , then .
- If , then .
Either way, we still get one negative and one positive fraction. Therefore, we can drop the absolute value signs.
Either
or
And, finally
People generally shorthand these two separate solutions as
This is known as The Quadratic Formula.
Let’s apply this to the previous three examples
Solve
To use the Quadratic Formula the quadratic equation has to have everything on one side and on the other. Our equation is already in that form.
Apply the Quadratic Formula
- Matching the leading terms tells us that .
- Matching the linear terms tells us that .
- Matching the constant terms tells us that .
These are the substitutions we need to perform in the Quadratic Formula.
Either or
Either or
Either or
In each of the examples above the number of solutions was determined by . The inside of the square root, , is called the discriminant and its sign tells us how many real solutions the equation has.
- If , then there are two distinct real solutions.
- If , then there is one real solutions.
- If , then there are no distinct real solutions.
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more examples can be found by following this link
More Examples of Quadratics