If $f(x) = 2x^2 + 3x - 1$ And $g(x) = 2(x - 2)^2$, What Is The Equivalent Form Of $f(x) + G(x$\]?A. $4x^2 - 5x + 7$ B. $4x^2 + 3x + 7$ C. $6x^2 - 13x + 15$ D. $6x^2 + 3x + 15$

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. In this article, we will explore how to simplify two given functions, $f(x)$ and $g(x)$, and find their equivalent form when added together.

Understanding the Functions

Function $f(x)$

The function $f(x)$ is defined as $f(x) = 2x^2 + 3x - 1$. This is a quadratic function, which means it has a squared term ($x^2$) and a linear term ($3x$).

Function $g(x)$

The function $g(x)$ is defined as $g(x) = 2(x - 2)^2$. This is also a quadratic function, but it has a squared term with a binomial ($x - 2$) inside.

Simplifying $g(x)$

To simplify $g(x)$, we need to expand the squared binomial using the formula $(a - b)^2 = a^2 - 2ab + b^2$.

g(x) = 2(x - 2)^2
g(x) = 2(x^2 - 2x + 4)
g(x) = 2x^2 - 4x + 8

Adding $f(x)$ and $g(x)$

Now that we have simplified $g(x)$, we can add it to $f(x)$ to find their equivalent form.

f(x) + g(x) = (2x^2 + 3x - 1) + (2x^2 - 4x + 8)
f(x) + g(x) = 4x^2 - x + 7

Comparing the Results

The equivalent form of $f(x) + g(x)$ is $4x^2 - x + 7$. Let's compare this result with the given options:

• A. $4x^2 - 5x + 7$
• B. $4x^2 + 3x + 7$
• C. $6x^2 - 13x + 15$
• D. $6x^2 + 3x + 15$

Our result, $4x^2 - x + 7$, does not match any of the given options. However, we can see that option A is close, but it has a different constant term.

Conclusion

In this article, we simplified two given functions, $f(x)$ and $g(x)$, and found their equivalent form when added together. We used algebraic techniques, such as expanding squared binomials and combining like terms, to simplify the expressions. Our result, $4x^2 - x + 7$, does not match any of the given options, but it provides a clear understanding of how to simplify algebraic expressions.

Final Answer

Introduction

In our previous article, we explored how to simplify two given functions, $f(x)$ and $g(x)$, and find their equivalent form when added together. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q: What is the difference between simplifying and solving an equation?

A: Simplifying an equation involves combining like terms and removing any unnecessary operations, while solving an equation involves finding the value of the variable that makes the equation true.

Q: How do I simplify a fraction with variables in the numerator and denominator?

A: To simplify a fraction with variables in the numerator and denominator, you need to find the greatest common factor (GCF) of the variables and cancel it out. For example, if you have the fraction $\frac{2x^2}{4x}$, you can simplify it by canceling out the $2x$ in the numerator and denominator.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables by combining like terms and removing any unnecessary operations. For example, if you have the expression $2x^2 + 3y^2 + 4x + 5y$, you can simplify it by combining the like terms $2x^2 + 4x$ and $3y^2 + 5y$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to follow the rules of exponentiation. For example, if you have the expression $2^3 \cdot 2^2$, you can simplify it by combining the exponents using the rule $a^m \cdot a^n = a^{m+n}$.

Q: Can I simplify an expression with absolute values?

A: Yes, you can simplify an expression with absolute values by removing the absolute value signs and considering both the positive and negative cases. For example, if you have the expression $|x - 2|$, you can simplify it by considering both the cases $x - 2 \geq 0$ and $x - 2 < 0$.

Q: How do I simplify an expression with radicals?

A: To simplify an expression with radicals, you need to follow the rules of radical operations. For example, if you have the expression $\sqrt{16} + \sqrt{25}$, you can simplify it by evaluating the square roots and combining the results.

Q: Can I simplify an expression with trigonometric functions?

A: Yes, you can simplify an expression with trigonometric functions by using trigonometric identities and formulas. For example, if you have the expression $\sin^2 x + \cos^2 x$, you can simplify it using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We covered topics such as simplifying fractions, expressions with multiple variables, exponents, absolute values, radicals, and trigonometric functions. By following the rules and techniques outlined in this article, you can simplify complex algebraic expressions and solve equations with confidence.

Final Tips

• Always start by simplifying the expression inside the parentheses.
• Use the order of operations (PEMDAS) to simplify the expression.
• Combine like terms and remove any unnecessary operations.
• Use trigonometric identities and formulas to simplify expressions with trigonometric functions.
• Practice, practice, practice! The more you practice simplifying algebraic expressions, the more confident you will become.