Simplify The Expression: $\frac{4}{4x^2} - \frac{4}{12x+9}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems more efficiently. When dealing with fractions, combining them can be a bit challenging, but with the right approach, it can be a breeze. In this article, we will simplify the expression 44x2−412x+9\frac{4}{4x^2} - \frac{4}{12x+9} using a step-by-step approach.

Understanding the Expression

The given expression is a combination of two fractions: 44x2\frac{4}{4x^2} and 412x+9\frac{4}{12x+9}. To simplify this expression, we need to find a common denominator for both fractions. The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Finding the Common Denominator

To find the LCM of 4x24x^2 and 12x+912x+9, we need to factorize both expressions. 4x24x^2 can be factorized as 4x24x^2, and 12x+912x+9 cannot be factorized further.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expressions
expr1 = 4*x**2
expr2 = 12*x + 9

# Find the LCM of the expressions
lcm = sp.lcm(expr1, expr2)

print(lcm)

The LCM of 4x24x^2 and 12x+912x+9 is 12x2+36x12x^2+36x. Now that we have the common denominator, we can rewrite both fractions with the common denominator.

Rewriting the Fractions

To rewrite the fractions, we need to multiply the numerator and denominator of each fraction by the necessary factors to get the common denominator.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expressions
expr1 = 4/(4*x**2)
expr2 = 4/(12*x + 9)

# Rewrite the fractions with the common denominator
expr1_rewritten = (4*(12*x**2 + 36*x))/(4*(12*x**2 + 36*x))
expr2_rewritten = (4*(4*x**2))/(12*x**2 + 36*x)

print(expr1_rewritten)
print(expr2_rewritten)

Simplifying the Expression

Now that we have rewritten both fractions with the common denominator, we can simplify the expression by combining the fractions.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expressions
expr1_rewritten = (4*(12*x**2 + 36*x))/(4*(12*x**2 + 36*x))
expr2_rewritten = (4*(4*x**2))/(12*x**2 + 36*x)

# Simplify the expression
simplified_expr = sp.simplify(expr1_rewritten - expr2_rewritten)

print(simplified_expr)

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems more efficiently. By following the steps outlined in this article, we can simplify the expression 44x2−412x+9\frac{4}{4x^2} - \frac{4}{12x+9} using a step-by-step approach. We found the common denominator, rewrote the fractions, and simplified the expression to get the final result.

Final Answer

The final answer is 4x2−12x−912x2+36x\boxed{\frac{4x^2-12x-9}{12x^2+36x}}.

Additional Resources

For more information on simplifying expressions, check out the following resources:

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

FAQs

Q: What is the common denominator of two fractions? A: The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Q: How do I find the LCM of two expressions? A: You can use a calculator or a computer algebra system (CAS) to find the LCM of two expressions.

Q: How do I rewrite fractions with a common denominator? A: You need to multiply the numerator and denominator of each fraction by the necessary factors to get the common denominator.

Introduction

In our previous article, we simplified the expression 44x2−412x+9\frac{4}{4x^2} - \frac{4}{12x+9} using a step-by-step approach. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A Guide

Q: What is the common denominator of two fractions?

A: The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Q: How do I find the LCM of two expressions?

A: You can use a calculator or a computer algebra system (CAS) to find the LCM of two expressions. Alternatively, you can factorize both expressions and find the LCM by identifying the common factors.

Q: How do I rewrite fractions with a common denominator?

A: You need to multiply the numerator and denominator of each fraction by the necessary factors to get the common denominator.

Q: How do I simplify an expression?

A: You can use a step-by-step approach to simplify an expression, starting with finding the common denominator, rewriting the fractions, and combining the fractions.

Q: What is the difference between simplifying an expression and combining fractions?

A: Simplifying an expression involves finding the common denominator and rewriting the fractions, while combining fractions involves adding or subtracting fractions with the same denominator.

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

A: Yes, you can simplify an expression with variables in the denominator by following the same steps as before.

Q: How do I handle fractions with negative exponents?

A: When dealing with fractions with negative exponents, you can rewrite the fraction with a positive exponent by taking the reciprocal of the fraction.

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by following the same steps as before.

Q: How do I check my work when simplifying an expression?

A: You can check your work by plugging in values for the variables and evaluating the expression.

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

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

  • Not finding the common denominator
  • Not rewriting the fractions correctly
  • Not combining the fractions correctly
  • Not checking the work

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems more efficiently. By following the steps outlined in this article and answering the FAQs, you can simplify expressions with confidence.

Additional Resources

For more information on simplifying expressions, check out the following resources:

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

Practice Problems

Try simplifying the following expressions:

  • 2x2−2x+1\frac{2}{x^2} - \frac{2}{x+1}
  • 3x2+3x−1\frac{3}{x^2} + \frac{3}{x-1}
  • 4x2−4x−2\frac{4}{x^2} - \frac{4}{x-2}

Answer Key

  • 2x+2x2(x+1)\frac{2x+2}{x^2(x+1)}
  • 3x−3x2(x−1)\frac{3x-3}{x^2(x-1)}
  • 4x−8x2(x−2)\frac{4x-8}{x^2(x-2)}