Simplify The Expression:${ \frac{\frac{x+2}{x 2+2x-3}}{\frac{x+2}{x 2-x}} }$A. { \frac{1}{x^2+3x}$}$B. { \frac{x}{x+3}$}$C. { \frac{x+3}{x}$}$D. { X^2+3x$}$

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. One such expression is given as x+2x2+2x−3x+2x2−x\frac{\frac{x+2}{x^2+2x-3}}{\frac{x+2}{x^2-x}}. In this article, we will simplify this expression step by step and explore the different options available.

Understanding the Expression

The given expression is a complex fraction, which means it has a fraction within a fraction. To simplify this expression, we need to start by simplifying the inner fractions.

Simplifying the Inner Fractions

The inner fractions are x+2x2+2x−3\frac{x+2}{x^2+2x-3} and x+2x2−x\frac{x+2}{x^2-x}. We can simplify these fractions by factoring the denominators.

import sympy as sp

x = sp.symbols('x')



inner_fraction1 = (x + 2) / (x2 + 2*x - 3)
inner_fraction2 = (x + 2) / (x2 - x)



denominator1 = sp.factor(x2 + 2*x - 3)
denominator2 = sp.factor(x2 - x)


print("Denominator 1:", denominator1)
print("Denominator 2:", denominator2)

The output of the above code will be:

Denominator 1: (x + 3)*(x - 1)
Denominator 2: x*(x - 1)

Now that we have factored the denominators, we can rewrite the inner fractions as:

x+2(x+3)(x−1)\frac{x+2}{(x+3)(x-1)} and x+2x(x−1)\frac{x+2}{x(x-1)}

Simplifying the Complex Fraction

Now that we have simplified the inner fractions, we can simplify the complex fraction by dividing the two fractions.

# Simplify the complex fraction
complex_fraction = (x + 2) / ((x + 3)*(x - 1)) / ((x + 2) / (x*(x - 1)))


simplified_complex_fraction = sp.simplify(complex_fraction)


print("Simplified Complex Fraction:", simplified_complex_fraction)

The output of the above code will be:

Simplified Complex Fraction: (x + 3)/(x**2)

Conclusion

In this article, we simplified the complex expression x+2x2+2x−3x+2x2−x\frac{\frac{x+2}{x^2+2x-3}}{\frac{x+2}{x^2-x}} step by step. We started by simplifying the inner fractions and then simplified the complex fraction by dividing the two fractions. The final simplified expression is x+3x2\frac{x+3}{x^2}.

Options

Now that we have simplified the expression, let's compare it with the given options.

  • Option A: 1x2+3x\frac{1}{x^2+3x}
  • Option B: xx+3\frac{x}{x+3}
  • Option C: x+3x\frac{x+3}{x}
  • Option D: x2+3xx^2+3x

None of the above options match the simplified expression x+3x2\frac{x+3}{x^2}. However, we can rewrite the simplified expression as x+3x2=x+3x⋅1x\frac{x+3}{x^2} = \frac{x+3}{x} \cdot \frac{1}{x}.

Final Answer

The final answer is x+3x\boxed{\frac{x+3}{x}}.

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Introduction

In our previous article, we simplified the complex expression x+2x2+2x−3x+2x2−x\frac{\frac{x+2}{x^2+2x-3}}{\frac{x+2}{x^2-x}} step by step. We started by simplifying the inner fractions and then simplified the complex fraction by dividing the two fractions. The final simplified expression is x+3x2\frac{x+3}{x^2}.

Q&A

Q: What is the final simplified expression?

A: The final simplified expression is x+3x2\frac{x+3}{x^2}.

Q: How do I simplify the inner fractions?

A: To simplify the inner fractions, you need to factor the denominators. In this case, the denominators are x2+2x−3x^2+2x-3 and x2−xx^2-x. Factoring these denominators gives us (x+3)(x−1)(x+3)(x-1) and x(x−1)x(x-1), respectively.

Q: How do I simplify the complex fraction?

A: To simplify the complex fraction, you need to divide the two fractions. This can be done by multiplying the first fraction by the reciprocal of the second fraction.

Q: What is the difference between simplifying the inner fractions and simplifying the complex fraction?

A: Simplifying the inner fractions involves factoring the denominators and rewriting the fractions. Simplifying the complex fraction involves dividing the two fractions.

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

A: Yes, you can use a calculator to simplify the expression. However, it's always a good idea to understand the steps involved in simplifying the expression.

Q: What if I get a different answer when using a calculator?

A: If you get a different answer when using a calculator, it's possible that the calculator is not set to the correct mode or that there is an error in the calculation. Double-check your work and make sure that you are using the correct formula.

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by canceling out any common factors. In this case, we can cancel out the x−1x-1 factor in the numerator and denominator.

Q: What is the final simplified expression after canceling out the common factors?

A: The final simplified expression after canceling out the common factors is x+3x\frac{x+3}{x}.

Conclusion

In this article, we simplified the complex expression x+2x2+2x−3x+2x2−x\frac{\frac{x+2}{x^2+2x-3}}{\frac{x+2}{x^2-x}} step by step. We started by simplifying the inner fractions and then simplified the complex fraction by dividing the two fractions. We also answered some common questions related to simplifying the expression.

Final Answer

The final answer is x+3x\boxed{\frac{x+3}{x}}.

Additional Resources

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