
Using the Quadratic Formula
This is a refresher section on using the quadratic formula to solve quadratic equations. Additionally, you will be introduced to the discriminant and how to use it to classify the number and type of solutions to a quadratic equation. This analysis is an important step in learning how to analyze the behavior of functions using algebraic and graphical methods.
The fourth method of solving a quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.
We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by and obtain a positive
. Given
,
, we will complete the square as follows:
1. First, move the constant term to the right side of the equal sign:
2. As we want the leading coefficient to equal , divide through by
:
3. Then, find of the middle term, and add
to both sides of the equal sign:
4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:
5. Now, use the square root property, which gives
6. Finally, add to both sides of the equation and combine the terms on the right side. Thus,
The Quadratic Formula
Written in standard form, , any quadratic equation can be solved using the quadratic formula:
HOW TO
Given a quadratic equation, solve it using the quadratic formula
Example 9
Solve the Quadratic Equation Using the Quadratic Formula
Solve the quadratic equation: .
Solution
Identify the coefficients: . Then use the quadratic formula.
Example 10
Solving a Quadratic Equation with the Quadratic Formula
Use the quadratic formula to solve .
Solution
First, we identify the coefficients: , and
.
Substitute these values into the quadratic formula.
Source: Rice University, https://openstax.org/books/college-algebra/pages/2-5-quadratic-equations
This work is licensed under a Creative Commons Attribution 4.0 License.