Factorize $3x^2 + 13x - 10$. Hence, Simplify $\frac{3x^2 + 13x - 10}{9x^2 - 4}$.
Introduction
In mathematics, factorization is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. This process is essential in solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will focus on factorizing the quadratic expression $3x^2 + 13x - 10$ and then simplify the resulting expression when divided by $9x^2 - 4$.
Factorizing Quadratic Expressions
A quadratic expression is a polynomial of degree two, which means it has the general form $ax^2 + bx + c$. To factorize a quadratic expression, we need to find two binomials whose product equals the original expression. The general form of a quadratic expression can be factored as:
where $m$, $n$, $p$, and $q$ are constants to be determined.
Factorizing $3x^2 + 13x - 10$
To factorize $3x^2 + 13x - 10$, we need to find two binomials whose product equals the original expression. We can start by looking for two numbers whose product is $-30$ (the product of the constant term and the coefficient of the quadratic term) and whose sum is $13$ (the coefficient of the linear term).
After some trial and error, we find that the numbers $15$ and $-2$ satisfy these conditions. Therefore, we can write:
This is the factored form of the quadratic expression $3x^2 + 13x - 10$.
Simplifying $\frac{3x^2 + 13x - 10}{9x^2 - 4}$
Now that we have factored the quadratic expression $3x^2 + 13x - 10$, we can simplify the resulting expression when divided by $9x^2 - 4$.
To simplify the expression, we can use the factored form of the numerator and the denominator:
We can then cancel out any common factors between the numerator and the denominator. In this case, we can cancel out the factor $x + 5$ from the numerator and the denominator:
However, we can further simplify the expression by factoring the denominator:
Therefore, we can write:
We can then cancel out the factor $3x - 2$ from the numerator and the denominator:
This is the simplified form of the expression $\frac{3x^2 + 13x - 10}{9x^2 - 4}$.
Conclusion
In this article, we have factorized the quadratic expression $3x^2 + 13x - 10$ and simplified the resulting expression when divided by $9x^2 - 4$. We have used the factored form of the numerator and the denominator to cancel out any common factors and simplify the expression. The final simplified form of the expression is $\frac{1}{3x + 2}$.
Final Answer
The final answer is:
Introduction
In our previous article, we factorized the quadratic expression $3x^2 + 13x - 10$ and simplified the resulting expression when divided by $9x^2 - 4$. In this article, we will answer some frequently asked questions related to the topic.
Q&A
Q: What is the process of factorizing a quadratic expression?
A: The process of factorizing a quadratic expression involves expressing it as a product of simpler expressions. This is done by finding two binomials whose product equals the original expression.
Q: How do I factorize a quadratic expression?
A: To factorize a quadratic expression, you need to find two numbers whose product is the product of the constant term and the coefficient of the quadratic term, and whose sum is the coefficient of the linear term. You can then write the quadratic expression as a product of these two binomials.
Q: What is the difference between factoring and simplifying an expression?
A: Factoring an expression involves expressing it as a product of simpler expressions, while simplifying an expression involves reducing it to its simplest form by canceling out any common factors.
Q: Can I simplify an expression by canceling out any common factors?
A: Yes, you can simplify an expression by canceling out any common factors between the numerator and the denominator.
Q: What is the final simplified form of the expression $\frac{3x^2 + 13x - 10}{9x^2 - 4}$?
A: The final simplified form of the expression $\frac{3x^2 + 13x - 10}{9x^2 - 4}$ is $\frac{1}{3x + 2}$.
Q: How do I know if an expression can be simplified?
A: You can simplify an expression if there are any common factors between the numerator and the denominator.
Q: Can I factorize a quadratic expression with a negative leading coefficient?
A: Yes, you can factorize a quadratic expression with a negative leading coefficient. The process is the same as factorizing a quadratic expression with a positive leading coefficient.
Q: What is the importance of factorizing and simplifying expressions?
A: Factorizing and simplifying expressions are important in mathematics because they help us to understand the properties of functions and to solve equations.
Conclusion
In this article, we have answered some frequently asked questions related to the topic of factorizing and simplifying expressions. We have discussed the process of factorizing a quadratic expression, the difference between factoring and simplifying an expression, and the importance of factorizing and simplifying expressions.
Final Answer
The final answer is:
Additional Resources
- Factorizing Quadratic Expressions: A Step-by-Step Guide
- Simplifying Expressions: A Guide to Canceling Out Common Factors
- The Importance of Factorizing and Simplifying Expressions in Mathematics
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