Solving Rational Equations (Easy Method)

Summary

In this video, viewers learn how to solve rational equations efficiently using factoring and clearing denominators. The instructor demonstrates this process through three examples, emphasizing the importance of factoring denominators, finding a common denominator, and multiplying both sides of the equation by it to eliminate fractions. After simplifying the equation, the instructor solves it like a regular algebraic equation and highlights the necessity of checking for extraneous solutions. By following these steps, viewers can solve rational equations with confidence and avoid common pitfalls such as division by zero.

Takeaways

• 😀 Factor denominators of rational equations to make solving easier.
• 😀 Look for a common denominator across all fractions before proceeding.
• 😀 Always multiply the numerator and denominator by the same factor to maintain equality.
• 😀 Clearing denominators simplifies the equation by eliminating fractions.
• 😀 Once the denominators are cleared, the equation becomes a simpler algebraic equation.
• 😀 Always check for extraneous solutions that might make the denominator zero.
• 😀 If a solution makes the denominator zero, it's considered a false solution and should be discarded.
• 😀 When factoring denominators, pay attention to special cases like the difference of squares.
• 😀 After clearing denominators, focus on solving the simplified equation like a regular algebra problem.
• 😀 If a quadratic equation results, be prepared to use the quadratic formula or other methods to solve it.
• 😀 Always ensure that solutions do not result in division by zero, which is undefined.

Q & A

What is the first step in solving rational equations according to the video?

The first step is to factor the denominators of all the fractions involved in the equation.

Why is factoring important when solving rational equations?

Factoring helps identify the components of the denominator, making it easier to find a common denominator and simplify the equation.

What should you do if a denominator has missing factors compared to others?

You need to multiply both the numerator and denominator of the fraction by the missing factors to create a common denominator.

How does multiplying the equation by the common denominator help?

Multiplying the entire equation by the common denominator clears the fractions by canceling out the denominators, simplifying the equation.

What happens after you clear the denominators in a rational equation?

Once the denominators are cleared, you are left with a simpler equation that can be solved algebraically, either by simplifying or applying other solving methods.

What method is used to solve quadratic equations in the examples?

The quadratic formula is used to solve the quadratic equation when factoring isn't possible or practical.

What is an extraneous solution, and why is it important to check for one?

An extraneous solution is a solution that makes the denominator equal to zero. It is important to check for these solutions because division by zero is undefined.

In the second example, what common denominator was used to clear the fractions?

The common denominator used was (x + 2)(x - 2).

How does one check if a solution is extraneous after solving the equation?

To check for extraneous solutions, substitute the solution back into the original equation to ensure it does not make any denominator equal to zero.

What is the significance of the quadratic formula in solving rational equations?

The quadratic formula is significant when the resulting equation is quadratic, and factoring is not possible. It provides a way to find the roots of the equation.