Solving Rational Equations

This page provides a comprehensive guide on solving równania wymierne (rational equations). It presents several examples with detailed solutions, demonstrating the step-by-step process for tackling these mathematical problems.

The first example shown is:

(4x(x + 2)) / ((2x+1)(x+3)) = 0

The solution process begins by determining the domain of the equation. In this case, the domain is all real numbers except -3, as x cannot equal -3 due to it making the denominator zero.

Definition: The domain of a rational equation is all real numbers except those that make the denominator equal to zero.

Next, the equation is solved by setting each factor in the numerator to zero:

4x = 0, which gives x = 0 x + 2 = 0, which gives x = -2

Example: When solving (4x(x + 2)) / ((2x+1)(x+3)) = 0, we set 4x = 0 and x + 2 = 0 separately.

The page continues with more complex examples, such as:

(x² + 7)(x-5)³ / (x³+1)(x+2) = 0

This equation requires careful consideration of the domain and involves solving multiple factors set to zero.

Highlight: When solving rational equations, always check if the solutions are within the domain of the equation.

The document also includes examples with linear factors in both numerator and denominator, demonstrating how to handle cases where potential solutions may not be in the domain of the equation.

Vocabulary: Równania wymierne zadania z rozwiązaniami refers to rational equation problems with solutions, which this document effectively demonstrates.

Throughout the page, there's an emphasis on the systematic approach to solving these equations, making it an excellent resource for students preparing for exams or looking to improve their skills in równania wymierne - zadania pdf (rational equations - exercises pdf).