How do you eliminate a square root in an equation?
Square both sides of the equation.
Why is it important to check for extraneous solutions in radical equations?
Squaring both sides of an equation can introduce solutions that do not satisfy the original equation.
What should you do after finding the LCD in a rational equation?
Multiply every term on both sides of the equation by the LCD to eliminate the fractions.
What should you check for when verifying solutions to rational equations?
Ensure that the solutions do not make any of the denominators in the original equation equal to zero.
How do you solve an absolute value equation?
Isolate the absolute value, then consider two cases: the expression inside the absolute value equals the positive value and the negative value on the other side of the equation. Solve each case separately.
What is the first step in solving $|2x + 1| = 5$?
Recognize the absolute value is already isolated. Proceed by setting up two equations: $2x + 1 = 5$ and $2x + 1 = -5$.
Explain how to solve a rational equation with multiple fractions.
Find the LCD of all fractions, multiply each term by the LCD, simplify, and solve the resulting equation. Always check for extraneous solutions.
How do you verify solutions in an absolute value equation?
Substitute each solution back into the original absolute value equation and check if the equation holds true.
Describe the process of solving $\sqrt{x} + 3 = 7$.
First, isolate the radical: $\sqrt{x} = 4$. Then, square both sides: $x = 16$. Finally, verify the solution in the original equation.
How do you handle extraneous solutions in rational equations?
Discard any solution that makes the denominator of any fraction in the original equation equal to zero.
What is a radical equation?
An equation containing a variable within a radical (e.g., square root, cube root).
What is an extraneous solution?
A solution obtained algebraically that does not satisfy the original equation.
What is a rational equation?
An equation containing variables in the denominators of fractions.
What is the Least Common Denominator (LCD)?
The smallest multiple that is common to all denominators in a set of fractions.
What is an absolute value equation?
An equation containing a variable within absolute value symbols.
Define inverse operations.
Operations that undo each other (e.g., addition and subtraction, multiplication and division).
What does it mean to isolate a radical?
To get the radical term by itself on one side of the equation.
What is the quadratic formula?
The formula used to find the solutions to any quadratic equation of the form $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
What is the absolute value of a number?
The distance of a number from zero; always non-negative.
What is the first step in solving a radical equation?
Isolate the radical term on one side of the equation.
