Simplify Fully: 4 X 2 + 4 X 2 X 2 − 2 \frac{4x^2 + 4x}{2x^2 - 2} 2 X 2 − 2 4 X 2 + 4 X ​

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the process of simplifying fractions, especially when dealing with quadratic expressions. In this article, we will simplify the given expression $\frac{4x^2 + 4x}{2x^2 - 2}$ by factoring and canceling common factors.

Understanding the Expression

The given expression is a fraction with a quadratic expression in the numerator and a quadratic expression in the denominator. To simplify this expression, we need to factor both the numerator and the denominator.

Factoring the Numerator

The numerator is $4x^2 + 4x$. We can factor out the greatest common factor (GCF) of the two terms, which is $4x$. Factoring out $4x$ gives us:

$$4x^2 + 4x = 4x(x + 1)$$

Factoring the Denominator

The denominator is $2x^2 - 2$. We can factor out the greatest common factor (GCF) of the two terms, which is $2$. Factoring out $2$ gives us:

$$2x^2 - 2 = 2(x^2 - 1)$$

Simplifying the Expression

Now that we have factored both the numerator and the denominator, we can simplify the expression by canceling common factors. The numerator is $4x(x + 1)$ and the denominator is $2(x^2 - 1)$. We can cancel out the common factor of $2$ from the numerator and the denominator:

$$\frac{4x(x + 1)}{2(x^2 - 1)} = \frac{2x(x + 1)}{x^2 - 1}$$

However, we can simplify further by factoring the denominator as a difference of squares:

$$x^2 - 1 = (x + 1)(x - 1)$$

Now we can cancel out the common factor of $(x + 1)$ from the numerator and the denominator:

$$\frac{2x(x + 1)}{(x + 1)(x - 1)} = \frac{2x}{x - 1}$$

Conclusion

In this article, we simplified the given expression $\frac{4x^2 + 4x}{2x^2 - 2}$ by factoring and canceling common factors. We started by factoring the numerator and the denominator, and then we canceled out common factors to simplify the expression. The final simplified expression is $\frac{2x}{x - 1}$.

Final Answer

The final answer is $\boxed{\frac{2x}{x - 1}}$.

Tips and Tricks

• When simplifying fractions, always look for common factors to cancel out.
• Factoring is a crucial step in simplifying expressions, especially when dealing with quadratic expressions.
• Be careful when canceling common factors, as it's easy to make mistakes.

Common Mistakes

• Not factoring the numerator and denominator properly.
• Canceling out common factors without checking if they are actually present.
• Not simplifying the expression further by factoring the denominator as a difference of squares.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

Further Reading

If you want to learn more about simplifying algebraic expressions, I recommend checking out the following resources:

• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the process of simplifying fractions, especially when dealing with quadratic expressions. In this article, we simplified the given expression $\frac{4x^2 + 4x}{2x^2 - 2}$ by factoring and canceling common factors. The final simplified expression is $\frac{2x}{x - 1}$.

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Introduction

In our previous article, we simplified the given expression $\frac{4x^2 + 4x}{2x^2 - 2}$ by factoring and canceling common factors. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor the numerator and denominator, if possible.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. For example, to factor $x^2 + 5x + 6$, you need to find two numbers whose product is $6$ and whose sum is $5$. The two numbers are $2$ and $3$, so the factored form is $(x + 2)(x + 3)$.

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to factor expressions of the form $x^2 - y^2$.

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

A: To simplify a fraction with a quadratic expression in the numerator and denominator, you need to factor both the numerator and denominator, and then cancel out common factors.

Q: What is the final simplified expression for $\frac{4x^2 + 4x}{2x^2 - 2}$?

A: The final simplified expression for $\frac{4x^2 + 4x}{2x^2 - 2}$ is $\frac{2x}{x - 1}$.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. However, you need to be careful not to divide by zero.

Q: How do I know if an expression is already simplified?

A: An expression is already simplified if there are no common factors that can be canceled out.

Q: Can I use a calculator to simplify an expression?

A: Yes, you can use a calculator to simplify an expression. However, it's always a good idea to check your work by hand to make sure you understand the process.

Tips and Tricks

• Always factor the numerator and denominator, if possible.
• Use the difference of squares formula to factor expressions of the form $x^2 - y^2$.
• Be careful not to divide by zero.
• Check your work by hand to make sure you understand the process.

Common Mistakes

• Not factoring the numerator and denominator properly.
• Canceling out common factors without checking if they are actually present.
• Not simplifying the expression further by factoring the denominator as a difference of squares.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

Further Reading

If you want to learn more about simplifying algebraic expressions, I recommend checking out the following resources:

• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the process of simplifying fractions, especially when dealing with quadratic expressions. In this article, we answered some frequently asked questions (FAQs) related to simplifying algebraic expressions. We hope this article has been helpful in understanding the process of simplifying algebraic expressions.