Radical Equation

Definition

A radical equation is an algebraic equation that contains one or more square root terms. Solving radical equations involves isolating the radical expression and then using algebraic techniques to find the values of the variable that satisfy the equation.

5 Must Know Facts For Your Next Test

1. Radical equations can have one or more solutions, and it is important to check for extraneous solutions.
2. Solving radical equations often involves isolating the radical expression and then using the property of squares to eliminate the radical.
3. When solving radical equations, it is crucial to check the final solution by substituting it back into the original equation.
4. The process of solving radical equations may involve multiple steps, such as simplifying, isolating the radical, and then squaring both sides.
5. Radical equations can arise in various applications, such as geometry, physics, and real-world problem-solving.

Review Questions

• Explain the steps involved in solving a basic radical equation, such as $\sqrt{x} = 5$.
  • To solve the radical equation $\sqrt{x} = 5$, the first step is to isolate the radical expression by itself on one side of the equation. In this case, we can simply square both sides to eliminate the square root: $\sqrt{x} = 5 \implies x = 5^2 = 25$. The final solution is $x = 25$. It is important to check this solution by substituting it back into the original equation to ensure it satisfies the equation.
• Describe the process of identifying and handling extraneous solutions when solving radical equations.
  • When solving radical equations, it is possible to obtain extraneous solutions, which are values of the variable that satisfy the transformed equation but not the original problem statement. To identify and handle extraneous solutions, one must carefully check the final solution by substituting it back into the original equation. If the solution does not satisfy the original equation, then it is an extraneous solution and should be discarded. Recognizing and addressing extraneous solutions is a crucial step in solving radical equations accurately.
• Explain how the property of squares can be used to simplify and solve more complex radical equations, such as $\sqrt{x + 2} = \sqrt{x} + 3$.
  • For the radical equation $\sqrt{x + 2} = \sqrt{x} + 3$, we can use the property of squares to solve it. First, we square both sides of the equation: $x + 2 = \left(\sqrt{x} + 3\right)^2 = x + 6\sqrt{x} + 9$. Next, we subtract $x$ from both sides to isolate the radical expression: $2 = 6\sqrt{x} + 9$. Then, we subtract 9 from both sides and divide by 6 to solve for $\sqrt{x}$: $\sqrt{x} = -1$. Finally, we square both sides to find the value of $x$: $x = 1$. This demonstrates how the property of squares can be used to simplify and solve more complex radical equations.

Related terms

Radical Expression: A radical expression is a mathematical expression that contains one or more square root, cube root, or other root symbols.

Extraneous Solution: An extraneous solution is a value of the variable that satisfies the original radical equation but not the original problem statement.

Squaring Both Sides: A technique used to solve radical equations by raising both sides of the equation to the power of 2 to eliminate the square root term.