Solve $\frac{x^2 - 3x + 4}{2x - 8} = \frac{x}{2}$.

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Introduction

Solving rational equations is a crucial aspect of algebra, and it requires a deep understanding of fractions, exponents, and algebraic manipulations. In this article, we will focus on solving the rational equation x2−3x+42x−8=x2\frac{x^2 - 3x + 4}{2x - 8} = \frac{x}{2}, which involves simplifying and manipulating fractions to isolate the variable. We will use various techniques, including factoring, canceling, and cross-multiplication, to solve this equation.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by canceling out any common factors in the numerator and denominator. In this case, we can factor the numerator as (x−4)(x−1)(x - 4)(x - 1) and the denominator as 2(x−4)2(x - 4). We can then cancel out the common factor (x−4)(x - 4) from both the numerator and denominator.

import sympy as sp

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

# Define the equation
eq = (x**2 - 3*x + 4) / (2*x - 8) - x / 2

# Simplify the equation
simplified_eq = sp.simplify(eq)
print(simplified_eq)

Step 2: Cross-Multiply

After simplifying the equation, we can cross-multiply to eliminate the fractions. This involves multiplying both sides of the equation by the denominator of the fraction.

# Cross-multiply
cross_multiplied_eq = sp.Eq(simplified_eq * (2*x - 8), 0)
print(cross_multiplied_eq)

Step 3: Expand and Simplify

After cross-multiplying, we can expand and simplify the equation to isolate the variable.

# Expand and simplify
expanded_eq = sp.expand(cross_multiplied_eq)
simplified_eq = sp.simplify(expanded_eq)
print(simplified_eq)

Step 4: Factor and Solve

Finally, we can factor the equation and solve for the variable.

# Factor and solve
factored_eq = sp.factor(simplified_eq)
solution = sp.solve(factored_eq, x)
print(solution)

Conclusion

In this article, we have solved the rational equation x2−3x+42x−8=x2\frac{x^2 - 3x + 4}{2x - 8} = \frac{x}{2} using various techniques, including simplifying, cross-multiplying, expanding, and factoring. We have used Python code to demonstrate each step of the solution process. The final solution is x=4x = 4.

Final Answer

The final answer is 4\boxed{4}.

Related Topics

  • Solving rational equations
  • Factoring and canceling
  • Cross-multiplication
  • Expanding and simplifying
  • Factoring and solving

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Python for Data Analysis" by Wes McKinney

Code

import sympy as sp

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

# Define the equation
eq = (x**2 - 3*x + 4) / (2*x - 8) - x / 2

# Simplify the equation
simplified_eq = sp.simplify(eq)

# Cross-multiply
cross_multiplied_eq = sp.Eq(simplified_eq * (2*x - 8), 0)

# Expand and simplify
expanded_eq = sp.expand(cross_multiplied_eq)
simplified_eq = sp.simplify(expanded_eq)

# Factor and solve
factored_eq = sp.factor(simplified_eq)
solution = sp.solve(factored_eq, x)

print(solution)

Note: The code is provided as a reference and can be used to verify the solution process. However, it is not necessary to run the code to understand the solution.

Introduction

Solving rational equations can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will provide answers to some common questions related to solving rational equations, including factoring, canceling, cross-multiplication, expanding, and simplifying.

Q: What is a rational equation?

A: A rational equation is an equation that contains fractions, where the numerator and denominator are polynomials.

Q: How do I simplify a rational equation?

A: To simplify a rational equation, you can factor the numerator and denominator, cancel out any common factors, and then cross-multiply to eliminate the fractions.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to eliminate fractions in a rational equation. It involves multiplying both sides of the equation by the denominator of the fraction.

Q: How do I expand and simplify a rational equation?

A: To expand and simplify a rational equation, you can use the distributive property to multiply out the terms, and then combine like terms to simplify the equation.

Q: What is factoring?

A: Factoring is a technique used to simplify a rational equation by expressing the numerator and denominator as products of simpler polynomials.

Q: How do I factor a rational equation?

A: To factor a rational equation, you can look for common factors in the numerator and denominator, and then factor out those common factors.

Q: What is canceling?

A: Canceling is a technique used to simplify a rational equation by canceling out any common factors in the numerator and denominator.

Q: How do I cancel out common factors in a rational equation?

A: To cancel out common factors in a rational equation, you can look for common factors in the numerator and denominator, and then cancel out those common factors.

Q: What is the final step in solving a rational equation?

A: The final step in solving a rational equation is to solve for the variable by isolating it on one side of the equation.

Q: How do I solve for the variable in a rational equation?

A: To solve for the variable in a rational equation, you can use algebraic manipulations to isolate the variable on one side of the equation.

Q: What are some common mistakes to avoid when solving rational equations?

A: Some common mistakes to avoid when solving rational equations include:

  • Not factoring the numerator and denominator
  • Not canceling out common factors
  • Not cross-multiplying to eliminate fractions
  • Not expanding and simplifying the equation
  • Not solving for the variable

Conclusion

Solving rational equations can be a challenging task, but with the right techniques and strategies, it can be made easier. By understanding the concepts of factoring, canceling, cross-multiplication, expanding, and simplifying, you can solve rational equations with confidence.

Final Answer

The final answer is 4\boxed{4}.

Related Topics

  • Solving rational equations
  • Factoring and canceling
  • Cross-multiplication
  • Expanding and simplifying
  • Factoring and solving

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Python for Data Analysis" by Wes McKinney

Code

import sympy as sp

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

# Define the equation
eq = (x**2 - 3*x + 4) / (2*x - 8) - x / 2

# Simplify the equation
simplified_eq = sp.simplify(eq)

# Cross-multiply
cross_multiplied_eq = sp.Eq(simplified_eq * (2*x - 8), 0)

# Expand and simplify
expanded_eq = sp.expand(cross_multiplied_eq)
simplified_eq = sp.simplify(expanded_eq)

# Factor and solve
factored_eq = sp.factor(simplified_eq)
solution = sp.solve(factored_eq, x)

print(solution)

Note: The code is provided as a reference and can be used to verify the solution process. However, it is not necessary to run the code to understand the solution.