Introduction Completing the square can be a long process, and not all quadratic expressions can be factored. Rather than completing the square or factoring, we can use a formula that can be derived from the process of completing the square. This formula, called the quadratic formula, can be used to solve any quadratic equation in standard form, ax 2 + bx + c = 0. 1 5.2.4: Applying the Quadratic Formula
17
Embed
Introduction Completing the square can be a long process, and not all quadratic expressions can be factored. Rather than completing the square or factoring,
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
IntroductionCompleting the square can be a long process, and not all quadratic expressions can be factored. Rather than completing the square or factoring, we can use a formula that can be derived from the process of completing the square. This formula, called the quadratic formula, can be used to solve any quadratic equation in standard form, ax2 + bx + c = 0.
1
5.2.4: Applying the Quadratic Formula
Key Concepts• A quadratic equation in standard form, ax2 + bx + c = 0,
can be solved for x by using the quadratic formula:
• Solutions of quadratic equations are also called roots. • The expression under the radical, b2 – 4ac, is called the
discriminant. • The discriminant tells us the number and type of
solutions for the equation.
2
5.2.4: Applying the Quadratic Formula
Key Concepts, continued
3
5.2.4: Applying the Quadratic Formula
DiscriminantNumber and type
of solutions
Negative Two complex solutions
0 One real, rational solution
Positive and a perfect square Two real, rational solutions
Positive and not a perfect square
Two real, irrational solutions
Common Errors/Misconceptions• not setting the quadratic equation equal to 0 before
determining the values of a, b, and c • forgetting to use ± for problems with two solutions • forgetting to change the sign of b • dividing by a or by 2 instead of by 2a • not correctly following the order of operations • not fully simplifying solutions
4
5.2.4: Applying the Quadratic Formula
Guided Practice
Example 2Use the discriminant of 3x2 – 5x + 1 = 0 to identify the number and type of solutions.
5
5.2.4: Applying the Quadratic Formula
Guided Practice: Example 2, continued
1. Determine a, b, and c. a = 3, b = –5, and c = 1
6
5.2.4: Applying the Quadratic Formula
Guided Practice: Example 2, continued
2. Substitute the values for a, b, and c into the formula for the discriminant, b2 – 4ac.
b2 – 4ac = (–5)2 – 4(3)(1) = 25 – 12 = 13
The discriminant of 3x2 – 5x + 1 = 0 is 13.
7
5.2.4: Applying the Quadratic Formula
Guided Practice: Example 2, continued
3. Use what you know about the discriminant to determine the number and type of solutions for the quadratic equation. The discriminant, 13, is positive, but it is not a perfect square. Therefore, there will be two real, irrational solutions.