IndietroSolving Quadratic Equations: Study Notes and Examples
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Quadratic Functions and Equations
Solving Quadratic Equations
Quadratic equations are polynomial equations of degree two, generally written in the standard form:
Standard Form: , where
There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.
Factoring Method
Express the quadratic equation in the form .
Factor the quadratic expression, if possible.
Set each factor equal to zero and solve for .
Quadratic Formula
If factoring is not possible, use the quadratic formula:
Discriminant: determines the nature of the roots.
Examples
Example 1: Solve Solution:
Rewrite as
Factor:
Set each factor to zero: or
Solutions: or
Example 2: Solve Solution:
Rewrite as
Factor:
Set each factor to zero: or
Solutions: or
Key Properties
Quadratic equations can have two real solutions, one real solution, or two complex solutions depending on the discriminant.
Factoring is possible when the quadratic can be written as a product of two binomials.
Table: Nature of Roots Based on Discriminant
Discriminant () | Nature of Roots |
|---|---|
> 0 | Two distinct real roots |
= 0 | One real root (repeated) |
< 0 | Two complex roots |
Additional info: These examples and explanations cover the basic methods for solving quadratic equations, which are foundational for further study in trigonometry and algebra.
