Rational and Radical Equations Flashcard

Front

What is a rational equation?

Back

A rational equation is an equation that involves at least one rational expression, which is a fraction where the numerator and/or denominator are polynomials.

Front

What is a radical equation?

Back

A radical equation is an equation in which a variable is contained within a radical (square root, cube root, etc.).

Front

How do you solve a rational equation?

Back

To solve a rational equation, find a common denominator, eliminate the fractions by multiplying both sides by that denominator, and then solve the resulting equation.

Front

What is an extraneous solution?

Back

An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

How can you identify extraneous solutions in rational equations?

Back

To identify extraneous solutions, substitute the potential solutions back into the original equation to check if they hold true.

Front

What is the first step in solving radical equations?

Back

The first step in solving radical equations is to isolate the radical on one side of the equation.

Front

What is the importance of checking solutions in radical equations?

Back

Checking solutions in radical equations is important because squaring both sides can introduce extraneous solutions that do not satisfy the original equation.

Front

What is the process for solving the equation: \( \sqrt{x+5} = x-1 \)?

Back

To solve \( \sqrt{x+5} = x-1 \), square both sides to eliminate the radical, then solve the resulting equation and check for extraneous solutions.

Front

What is the common denominator for the rational equation \( \frac{2x}{x+3} = \frac{2}{x+5} + \frac{4}{(x+5)(x+3)} \)?

Back

The common denominator for the equation is \( (x+3)(x+5) \).

Front

What is the solution to the rational equation \( \frac{x^2}{x+2} = \frac{4}{x+2} + 1 \)?

Back

The solution to the equation is \( x = 3 \).