Which Shows The Factored Form Of $x^2 - 12x - 45$?A. $(x + 3)(x - 15)$B. \$(x - 3)(x - 15)$[/tex\]C. $(x + 3)(x + 15)$D. $(x - 3)(x + 15)$
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Introduction
Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve for unknown variables. In this article, we will explore the process of factoring quadratic expressions and apply it to a specific problem. We will examine the given quadratic expression $x^2 - 12x - 45$ and determine which of the provided options shows its factored form.
Understanding Quadratic Expressions
A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the general form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The quadratic expression can be factored into the product of two binomials, which is a polynomial of degree one.
Factoring Quadratic Expressions
To factor a quadratic expression, we need to find two numbers whose product is equal to the constant term ($c$) and whose sum is equal to the coefficient of the linear term ($b$). These two numbers are called the factors of the quadratic expression.
Step 1: Identify the Constant Term and the Coefficient of the Linear Term
In the given quadratic expression $x^2 - 12x - 45$, the constant term is $-45$, and the coefficient of the linear term is $-12$.
Step 2: Find the Factors of the Constant Term
We need to find two numbers whose product is equal to $-45$. These numbers are $-15$ and $3$, since $-15 \times 3 = -45$.
Step 3: Find the Factors of the Coefficient of the Linear Term
We need to find two numbers whose sum is equal to $-12$. These numbers are $-3$ and $-9$, since $-3 + (-9) = -12$.
Step 4: Write the Factored Form
Now that we have found the factors of the constant term and the coefficient of the linear term, we can write the factored form of the quadratic expression. We need to multiply the two binomials, where the first binomial has the factors of the constant term and the second binomial has the factors of the coefficient of the linear term.
Factoring the Given Quadratic Expression
Let's apply the steps we learned to factor the given quadratic expression $x^2 - 12x - 45$.
Step 1: Identify the Constant Term and the Coefficient of the Linear Term
The constant term is $-45$, and the coefficient of the linear term is $-12$.
Step 2: Find the Factors of the Constant Term
The factors of the constant term are $-15$ and $3$.
Step 3: Find the Factors of the Coefficient of the Linear Term
The factors of the coefficient of the linear term are $-3$ and $-9$.
Step 4: Write the Factored Form
Now that we have found the factors of the constant term and the coefficient of the linear term, we can write the factored form of the quadratic expression. We need to multiply the two binomials, where the first binomial has the factors of the constant term and the second binomial has the factors of the coefficient of the linear term.
The factored form of the quadratic expression $x^2 - 12x - 45$ is $(x + 3)(x - 15)$.
Conclusion
In this article, we learned how to factor quadratic expressions by identifying the constant term and the coefficient of the linear term, finding the factors of these terms, and writing the factored form. We applied this process to the given quadratic expression $x^2 - 12x - 45$ and determined that its factored form is $(x + 3)(x - 15)$.
Answer
The correct answer is:
- A. $(x + 3)(x - 15)$
This is the factored form of the quadratic expression $x^2 - 12x - 45$.
Final Thoughts
Factoring quadratic expressions is an essential skill in algebra that helps us simplify complex equations and solve for unknown variables. By following the steps we learned in this article, you can factor any quadratic expression and determine its factored form.
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Introduction
In our previous article, we explored the process of factoring quadratic expressions and applied it to a specific problem. We learned how to identify the constant term and the coefficient of the linear term, find the factors of these terms, and write the factored form. In this article, we will answer some frequently asked questions about factoring quadratic expressions.
Q&A
Q: What is the difference between factoring and simplifying a quadratic expression?
A: Factoring a quadratic expression involves expressing it as the product of two binomials, while simplifying a quadratic expression involves combining like terms to reduce its complexity.
Q: How do I determine the factored form of a quadratic expression?
A: To determine the factored form of a quadratic expression, you need to identify the constant term and the coefficient of the linear term, find the factors of these terms, and write the factored form.
Q: What are the steps to factor a quadratic expression?
A: The steps to factor a quadratic expression are:
- Identify the constant term and the coefficient of the linear term.
- Find the factors of the constant term.
- Find the factors of the coefficient of the linear term.
- Write the factored form by multiplying the two binomials.
Q: Can I factor a quadratic expression with a negative leading coefficient?
A: Yes, you can factor a quadratic expression with a negative leading coefficient. The process is the same as factoring a quadratic expression with a positive leading coefficient.
Q: How do I know if a quadratic expression can be factored?
A: A quadratic expression can be factored if it can be expressed as the product of two binomials. You can try to factor the expression by following the steps we learned earlier.
Q: Can I factor a quadratic expression with a variable in the denominator?
A: No, you cannot factor a quadratic expression with a variable in the denominator. The expression must be in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
Q: How do I factor a quadratic expression with a coefficient of 1?
A: If the coefficient of the quadratic term is 1, you can factor the expression by finding the factors of the constant term and the coefficient of the linear term.
Q: Can I factor a quadratic expression with a negative constant term?
A: Yes, you can factor a quadratic expression with a negative constant term. The process is the same as factoring a quadratic expression with a positive constant term.
Q: How do I factor a quadratic expression with a coefficient of -1?
A: If the coefficient of the quadratic term is -1, you can factor the expression by finding the factors of the constant term and the coefficient of the linear term.
Conclusion
In this article, we answered some frequently asked questions about factoring quadratic expressions. We covered topics such as the difference between factoring and simplifying a quadratic expression, the steps to factor a quadratic expression, and how to determine the factored form of a quadratic expression.
Final Thoughts
Factoring quadratic expressions is an essential skill in algebra that helps us simplify complex equations and solve for unknown variables. By following the steps we learned in this article and answering the questions we posed, you can become proficient in factoring quadratic expressions and apply this skill to a wide range of problems.
Additional Resources
- Factoring Quadratic Expressions: A Step-by-Step Guide
- Factoring Quadratic Expressions: A Guide to the Factored Form
- Factoring Quadratic Expressions: A Guide to the Steps
Answer Key
- Q: What is the difference between factoring and simplifying a quadratic expression?
- A: Factoring a quadratic expression involves expressing it as the product of two binomials, while simplifying a quadratic expression involves combining like terms to reduce its complexity.
- Q: How do I determine the factored form of a quadratic expression?
- A: To determine the factored form of a quadratic expression, you need to identify the constant term and the coefficient of the linear term, find the factors of these terms, and write the factored form.
- Q: What are the steps to factor a quadratic expression?
- A: The steps to factor a quadratic expression are:
- Identify the constant term and the coefficient of the linear term.
- Find the factors of the constant term.
- Find the factors of the coefficient of the linear term.
- Write the factored form by multiplying the two binomials.
- Q: Can I factor a quadratic expression with a negative leading coefficient?
- A: Yes, you can factor a quadratic expression with a negative leading coefficient. The process is the same as factoring a quadratic expression with a positive leading coefficient.
- Q: How do I know if a quadratic expression can be factored?
- A: A quadratic expression can be factored if it can be expressed as the product of two binomials. You can try to factor the expression by following the steps we learned earlier.
- Q: Can I factor a quadratic expression with a variable in the denominator?
- A: No, you cannot factor a quadratic expression with a variable in the denominator. The expression must be in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
- Q: How do I factor a quadratic expression with a coefficient of 1?
- A: If the coefficient of the quadratic term is 1, you can factor the expression by finding the factors of the constant term and the coefficient of the linear term.
- Q: Can I factor a quadratic expression with a negative constant term?
- A: Yes, you can factor a quadratic expression with a negative constant term. The process is the same as factoring a quadratic expression with a positive constant term.
- Q: How do I factor a quadratic expression with a coefficient of -1?
- A: If the coefficient of the quadratic term is -1, you can factor the expression by finding the factors of the constant term and the coefficient of the linear term.