Simplify $\frac{6x^2 + 20}{4}$.A. $\frac{3x^2 + 10}{2}$ B. $\frac{3x^2 + 20}{2}$ C. $6x + 5$ D. $x + 5$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the expression 6x2+204\frac{6x^2 + 20}{4} using various methods and techniques. We will also provide step-by-step solutions and explanations to help readers understand the process.

Understanding the Expression

The given expression is 6x2+204\frac{6x^2 + 20}{4}. To simplify this expression, we need to understand the rules of algebraic simplification. The expression consists of two terms: 6x26x^2 and 2020. The numerator is a quadratic expression, and the denominator is a constant.

Method 1: Factoring the Numerator

One way to simplify the expression is to factor the numerator. We can factor out the greatest common factor (GCF) of the two terms in the numerator.

\frac{6x^2 + 20}{4} = \frac{2(3x^2 + 10)}{4}

Now, we can simplify the expression by canceling out the common factor of 2 in the numerator and denominator.

\frac{2(3x^2 + 10)}{4} = \frac{3x^2 + 10}{2}

Method 2: Dividing the Numerator and Denominator

Another way to simplify the expression is to divide the numerator and denominator by their greatest common factor (GCF). In this case, the GCF of the numerator and denominator is 2.

\frac{6x^2 + 20}{4} = \frac{6x^2 + 20}{2 \times 2}

Now, we can simplify the expression by dividing the numerator and denominator by 2.

\frac{6x^2 + 20}{2 \times 2} = \frac{3x^2 + 10}{2}

Method 3: Using Algebraic Identities

We can also simplify the expression using algebraic identities. One such identity is the difference of squares identity: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b).

However, in this case, we don't have a difference of squares, but we can use the identity a2+b2=(a+b)2−2aba^2 + b^2 = (a + b)^2 - 2ab.

6x^2 + 20 = (3x)^2 + 20

Now, we can rewrite the expression using the identity.

\frac{6x^2 + 20}{4} = \frac{(3x)^2 + 20}{4}

However, this method is not as straightforward as the previous two methods, and it may not be the most efficient way to simplify the expression.

Conclusion

In conclusion, we have simplified the expression 6x2+204\frac{6x^2 + 20}{4} using three different methods: factoring the numerator, dividing the numerator and denominator, and using algebraic identities. The simplified expression is 3x2+102\frac{3x^2 + 10}{2}.

Answer

The correct answer is A. 3x2+102\frac{3x^2 + 10}{2}.

Discussion

This problem requires a good understanding of algebraic simplification techniques, including factoring, dividing, and using algebraic identities. The student should be able to identify the greatest common factor (GCF) of the numerator and denominator and simplify the expression accordingly.

Tips and Variations

  • To make the problem more challenging, you can add more terms to the numerator or denominator.
  • To make the problem easier, you can simplify the expression by canceling out common factors before applying the simplification techniques.
  • You can also use other algebraic identities, such as the sum and difference of cubes, to simplify the expression.

Practice Problems

  1. Simplify the expression 2x2+153\frac{2x^2 + 15}{3}.
  2. Simplify the expression 5x2−202\frac{5x^2 - 20}{2}.
  3. Simplify the expression x2+104\frac{x^2 + 10}{4}.

Answer Key

  1. 2x2+153=2x2+5×33=2x2+51=2x2+5\frac{2x^2 + 15}{3} = \frac{2x^2 + 5 \times 3}{3} = \frac{2x^2 + 5}{1} = 2x^2 + 5
  2. 5x2−202=5x2−4×52=5x2−202=5(x2−4)2=5(x+2)(x−2)2\frac{5x^2 - 20}{2} = \frac{5x^2 - 4 \times 5}{2} = \frac{5x^2 - 20}{2} = \frac{5(x^2 - 4)}{2} = \frac{5(x + 2)(x - 2)}{2}
  3. $\frac{x^2 + 10}{4} = \frac{x^2 + 10}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{10})(x - \sqrt{10})}{2 \times 2} = \frac{(x + \sqrt{
    Simplify 6x2+204\frac{6x^2 + 20}{4}: A Comprehensive Guide ===========================================================

Q&A: Simplify 6x2+204\frac{6x^2 + 20}{4}

Q: What is the simplified form of 6x2+204\frac{6x^2 + 20}{4}?

A: The simplified form of 6x2+204\frac{6x^2 + 20}{4} is 3x2+102\frac{3x^2 + 10}{2}.

Q: How do I simplify the expression 6x2+204\frac{6x^2 + 20}{4}?

A: There are several ways to simplify the expression 6x2+204\frac{6x^2 + 20}{4}. You can factor the numerator, divide the numerator and denominator, or use algebraic identities.

Q: What is the greatest common factor (GCF) of the numerator and denominator?

A: The greatest common factor (GCF) of the numerator and denominator is 2.

Q: How do I factor the numerator?

A: To factor the numerator, you can look for the greatest common factor (GCF) of the two terms in the numerator. In this case, the GCF is 2.

6x^2 + 20 = 2(3x^2 + 10)

Q: How do I divide the numerator and denominator?

A: To divide the numerator and denominator, you can divide both by their greatest common factor (GCF). In this case, the GCF is 2.

\frac{6x^2 + 20}{4} = \frac{2(3x^2 + 10)}{2 \times 2} = \frac{3x^2 + 10}{2}

Q: Can I use algebraic identities to simplify the expression?

A: Yes, you can use algebraic identities to simplify the expression. One such identity is the difference of squares identity: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b).

However, in this case, we don't have a difference of squares, but we can use the identity a2+b2=(a+b)2−2aba^2 + b^2 = (a + b)^2 - 2ab.

6x^2 + 20 = (3x)^2 + 20

Now, we can rewrite the expression using the identity.

\frac{6x^2 + 20}{4} = \frac{(3x)^2 + 20}{4}

However, this method is not as straightforward as the previous two methods, and it may not be the most efficient way to simplify the expression.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

  • Not factoring the numerator or denominator
  • Not canceling out common factors
  • Not using algebraic identities correctly
  • Not checking for errors in the simplification process

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working on problems and exercises that involve simplifying expressions. You can also use online resources, such as math websites and apps, to practice simplifying expressions.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

  • Algebra: Simplifying expressions is a crucial skill in algebra, and it's used to solve equations and inequalities.
  • Calculus: Simplifying expressions is used in calculus to find derivatives and integrals.
  • Physics: Simplifying expressions is used in physics to describe the motion of objects and the behavior of physical systems.
  • Engineering: Simplifying expressions is used in engineering to design and analyze complex systems.

Conclusion

In conclusion, simplifying expressions is a crucial skill in mathematics, and it's used in many real-world applications. By understanding the rules and techniques involved in simplifying expressions, you can become proficient in simplifying expressions and apply it to various fields.