Factor Completely. If Not Factorable, Enter Prime.$x^2 + 49 = $ $\square$

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Introduction

In mathematics, factoring is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill that helps in solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will focus on factoring a quadratic expression of the form $x^2 + 49$ and determine whether it is factorable or not.

Understanding the Expression

The given expression is $x^2 + 49$. This is a quadratic expression in the form of $ax^2 + bx + c$, where $a = 1$, $b = 0$, and $c = 49$. To factor this expression, we need to find two binomials whose product is equal to the given expression.

Factoring the Expression

To factor the expression $x^2 + 49$, we need to find two binomials whose product is equal to the given expression. We can start by looking for two numbers whose product is equal to $49$ and whose sum is equal to $0$. However, since the sum of the two numbers is $0$, we can conclude that the two numbers are $0$ and $0$. But $0$ multiplied by $0$ is $0$, not $49$. Therefore, we need to look for two binomials whose product is equal to $x^2 + 49$.

Using the Difference of Squares Formula

We can use the difference of squares formula to factor the expression $x^2 + 49$. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. In this case, we can rewrite the expression $x^2 + 49$ as $(x)^2 - (7i)^2$, where $i$ is the imaginary unit.

Applying the Difference of Squares Formula

Using the difference of squares formula, we can factor the expression $(x)^2 - (7i)^2$ as $(x + 7i)(x - 7i)$. Therefore, we can conclude that the factored form of the expression $x^2 + 49$ is $(x + 7i)(x - 7i)$.

Conclusion

In conclusion, the expression $x^2 + 49$ is not factorable over the real numbers. However, it can be factored over the complex numbers as $(x + 7i)(x - 7i)$. This demonstrates the importance of considering different number systems when working with algebraic expressions.

Final Answer

The final answer is $\boxed{(x + 7i)(x - 7i)}$.

Additional Information

• The expression $x^2 + 49$ is an example of an irreducible quadratic expression.
• The difference of squares formula is a powerful tool for factoring quadratic expressions.
• The complex numbers provide a rich and interesting number system that can be used to extend the real numbers.

Related Topics

• Factoring quadratic expressions
• Difference of squares formula
• Complex numbers
• Algebraic expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Number Theory" by George E. Andrews

Keywords

• Factoring
• Quadratic expressions
• Difference of squares formula
• Complex numbers
• Algebraic expressions
  

Introduction

In our previous article, we discussed how to factor a quadratic expression of the form $x^2 + 49$. We also introduced the concept of the difference of squares formula and its application in factoring quadratic expressions. In this article, we will provide a Q&A section to address some common questions and concerns related to factoring quadratic expressions.

Q&A

Q1: What is the difference of squares formula?

A1: The difference of squares formula is a mathematical formula that states that $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to factor quadratic expressions of the form $x^2 - c^2$.

Q2: How do I apply the difference of squares formula?

A2: To apply the difference of squares formula, you need to identify the values of $a$ and $b$ in the expression $x^2 - c^2$. Then, you can rewrite the expression as $(x + c)(x - c)$ and factor it accordingly.

Q3: What is the difference between factoring and simplifying an expression?

A3: Factoring an expression involves expressing it as a product of simpler expressions, while simplifying an expression involves reducing it to its simplest form. For example, the expression $x^2 + 2x + 1$ can be factored as $(x + 1)^2$, but it cannot be simplified further.

Q4: Can all quadratic expressions be factored?

A4: No, not all quadratic expressions can be factored. Some quadratic expressions are irreducible, meaning that they cannot be expressed as a product of simpler expressions.

Q5: What is the significance of the imaginary unit $i$ in factoring quadratic expressions?

A5: The imaginary unit $i$ is used to extend the real numbers and provide a rich and interesting number system. In factoring quadratic expressions, $i$ is used to introduce complex roots and provide a way to factor expressions that cannot be factored over the real numbers.

Q6: How do I determine whether a quadratic expression is factorable or not?

A6: To determine whether a quadratic expression is factorable or not, you need to check if it can be expressed as a product of simpler expressions. If it can be expressed as a product of simpler expressions, then it is factorable. Otherwise, it is not factorable.

Q7: What are some common mistakes to avoid when factoring quadratic expressions?

A7: Some common mistakes to avoid when factoring quadratic expressions include:

• Not identifying the values of $a$ and $b$ correctly
• Not applying the difference of squares formula correctly
• Not checking for complex roots
• Not simplifying the expression further

Conclusion

In conclusion, factoring quadratic expressions is a crucial skill in mathematics that involves expressing an algebraic expression as a product of simpler expressions. By understanding the difference of squares formula and its application, you can factor quadratic expressions with ease. Additionally, by avoiding common mistakes and understanding the significance of the imaginary unit $i$, you can become proficient in factoring quadratic expressions.

Final Answer

The final answer is $\boxed{(x + 7i)(x - 7i)}$.

Additional Information

• The difference of squares formula is a powerful tool for factoring quadratic expressions.
• The imaginary unit $i$ is used to extend the real numbers and provide a way to factor expressions that cannot be factored over the real numbers.
• Factoring quadratic expressions is a crucial skill in mathematics that involves expressing an algebraic expression as a product of simpler expressions.

Related Topics

• Factoring quadratic expressions
• Difference of squares formula
• Complex numbers
• Algebraic expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Number Theory" by George E. Andrews

Keywords

• Factoring
• Quadratic expressions
• Difference of squares formula
• Complex numbers
• Algebraic expressions