What is Quadratic Equation Factorization?
Quadratic equation factorization is the process of converting an equation from the form to the form , where and are the roots of the quadratic equation.
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Quadratic equation factorization is the process of converting an equation from the form to the form , where and are the roots of the quadratic equation.
Note that the roots of a quadratic equation are the values of that make the equation equal to zero. When we convert the equation to its factored form, we can easily find its roots.
A quadratic equation in standard form is written as:
where , , and are constants and .
Factorization is based on the following property: If a product equals zero, then at least one of its factors must equal zero.
This means if , then:
Here are the general steps to factor a quadratic equation :
Factoring the equation:
In this equation, , , and .
Step : The equation is already in standard form with the right side equal to zero.
Step : We need to find two numbers that:
Factors of are , , , and . Possible factor pairs are and . The pair gives a sum of , which matches the value of .
Step : We can write the equation as:
Step : From the factored form above, we get:
Therefore, the roots of the equation are and .
Factoring the equation:
In this equation, , , and .
Factorization When Coefficient
When the coefficient is not equal to , we need some modifications in the factorization steps. There are several approaches:
Method Using Factors of
If we know one of the roots of a quadratic equation, we can use this information to find the complete factorization.
Example: One of the roots of the equation is
If is a root of the equation, then is one of its factors.
We can substitute into the original equation:
Now we can write the equation as .
Using the factorization method, we factor it as:
The roots of the equation are and .
Form
For equations without a constant term, we can factor out directly:
The roots are and .
Example:
The roots are and .
Form
For equations without an term, we can use the difference of squares pattern:
Not all quadratic equations can be easily factored using rational numbers. In such cases, we can use the quadratic formula:
A quadratic equation can be factored with rational numbers if the discriminant is a perfect square.
Factor the following quadratic equations:
Step : Identify the coefficients
Step : Find two numbers that when multiplied give and when added give
Step : Factorization
Step : Determine the roots of the equation
Therefore, the roots of the equation are and .
Step : Identify the coefficients
Step : Identify the coefficients
Step : Identify the coefficients
Step : Identify as a difference of squares
Step : The equation is already in standard form with the right side equal to zero.
Step : We need to find two numbers that:
Factors of are pairs of numbers with opposite signs:
The pair gives a sum of , which matches the value of .
Step : We can write the equation as:
We can group the terms:
Step : From the factored form above, we get:
Therefore, the roots of the equation are and .
Example of Factorization:
In this equation, , , and .
Step : Calculate
Step : Find a pair of factors of that when added give :
Factors of :
The pair gives a sum of , which matches the value of .
Step : Split the term into :
Step : Factor by grouping:
Step : Determine the roots of the equation:
Therefore, the roots of the equation are and .
Example:
The roots are and .
Step : Find two numbers that when multiplied give and when added give
Step : Factorization
Step : Determine the roots of the equation
Therefore, the roots of the equation are and .
Step : Find two numbers that when multiplied give and when added give
Step : Factorization
Step : Determine the roots of the equation
Therefore, the roots of the equation are and .
Step : Find two numbers that when multiplied give and when added give
Step : Factorization
Step : Determine the roots of the equation
Therefore, the roots of the equation are and .
Step : Use the difference of squares formula
Step : Determine the roots of the equation
Therefore, the roots of the equation are and .