Which Of The Following Expressions Is Equal To $x^2 + 25$?A. $(x + 10i)(x - 15i)$ B. \$(x - 5i)(x + 5i)$[/tex\] C. $(x + 5i)^2$ D. $(x - 5i)^2$
Introduction
Quadratic expressions are a fundamental concept in mathematics, and they can be used to solve a wide range of problems in various fields. In this article, we will explore the concept of quadratic expressions and how to solve them using complex numbers. We will also examine the given expressions and determine which one is equal to $x^2 + 25$.
Understanding Quadratic Expressions
A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic expressions can be solved using various methods, including factoring, completing the square, and the quadratic formula.
Complex Numbers
Complex numbers are numbers that have both real and imaginary parts. They are used to extend the real number system and provide a way to solve equations that cannot be solved using real numbers alone. Complex numbers are denoted by $a + bi$, where $a$ is the real part and $b$ is the imaginary part.
Solving the Given Expressions
Now, let's examine the given expressions and determine which one is equal to $x^2 + 25$.
A. $(x + 10i)(x - 15i)$
To solve this expression, we can use the distributive property to expand it:
Since $i^2 = -1$, we can simplify the expression:
Combining like terms, we get:
This expression is not equal to $x^2 + 25$.
B. $(x - 5i)(x + 5i)$
To solve this expression, we can use the distributive property to expand it:
Since $i^2 = -1$, we can simplify the expression:
This expression is equal to $x^2 + 25$.
C. $(x + 5i)^2$
To solve this expression, we can use the formula $(a + b)^2 = a^2 + 2ab + b^2$:
Since $i^2 = -1$, we can simplify the expression:
This expression is not equal to $x^2 + 25$.
D. $(x - 5i)^2$
To solve this expression, we can use the formula $(a - b)^2 = a^2 - 2ab + b^2$:
Since $i^2 = -1$, we can simplify the expression:
This expression is not equal to $x^2 + 25$.
Conclusion
In conclusion, the expression $(x - 5i)(x + 5i)$ is equal to $x^2 + 25$. This expression can be solved using the distributive property and the formula for squaring a binomial. The other expressions do not equal $x^2 + 25$ and can be solved using similar methods.
Final Answer
Introduction
In our previous article, we explored the concept of quadratic expressions and how to solve them using complex numbers. We also examined the given expressions and determined which one is equal to $x^2 + 25$. In this article, we will provide a Q&A guide to help you better understand quadratic expressions with complex numbers.
Q&A Guide
Q: What is a quadratic expression?
A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.
Q: What is a complex number?
A: A complex number is a number that has both real and imaginary parts. It is denoted by $a + bi$, where $a$ is the real part and $b$ is the imaginary part.
Q: How do I solve a quadratic expression with complex numbers?
A: To solve a quadratic expression with complex numbers, you can use various methods, including factoring, completing the square, and the quadratic formula. You can also use the distributive property and the formula for squaring a binomial to expand and simplify the expression.
Q: What is the difference between a real number and a complex number?
A: A real number is a number that has no imaginary part, while a complex number is a number that has both real and imaginary parts. For example, $3$ is a real number, while $3 + 4i$ is a complex number.
Q: How do I simplify a complex number expression?
A: To simplify a complex number expression, you can use the following steps:
- Expand the expression using the distributive property.
- Combine like terms.
- Simplify the expression by canceling out any common factors.
Q: What is the formula for squaring a binomial?
A: The formula for squaring a binomial is $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$.
Q: How do I use the quadratic formula to solve a quadratic expression?
A: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula and simplify.
Q: What is the difference between a quadratic expression and a polynomial expression?
A: A quadratic expression is a polynomial of degree two, while a polynomial expression is a polynomial of any degree. For example, $x^2 + 3x + 4$ is a quadratic expression, while $x^3 + 2x^2 + 3x + 4$ is a polynomial expression.
Conclusion
In conclusion, quadratic expressions with complex numbers can be solved using various methods, including factoring, completing the square, and the quadratic formula. By understanding the concepts of quadratic expressions and complex numbers, you can better solve problems involving these topics.
Final Tips
- Make sure to simplify complex number expressions by combining like terms and canceling out any common factors.
- Use the distributive property and the formula for squaring a binomial to expand and simplify complex number expressions.
- Practice solving quadratic expressions with complex numbers to become more comfortable with the concepts.
Common Mistakes
- Failing to simplify complex number expressions by combining like terms and canceling out any common factors.
- Using the wrong formula for squaring a binomial.
- Not using the distributive property to expand complex number expressions.
Additional Resources
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equations
- Wolfram Alpha: Quadratic Equations
Final Answer
The final answer is B. $(x - 5i)(x + 5i)$.