Understanding Extraneous Solutions in Radical Equations

This video tutorial explains how to identify extraneous solutions when solving radical equations. It begins with an introduction to the concept, followed by a detailed walkthrough of solving a radical equation and identifying which solutions are extraneous. The tutorial includes two example problems, demonstrating the process of solving and verifying solutions to ensure they fit the original equation. The key takeaway is the importance of checking solutions in the original equation to determine if they are extraneous.

What is the first step in identifying extraneous solutions in a radical equation?

• Multiply both sides by a constant

• Plug solutions back into the original equation

• Add a constant to both sides

• Subtract a constant from both sides

When solving the equation \( \sqrt{x+2} + 4 = x \), what operation is performed after moving the constant to the other side?

• Taking the square root of both sides

• Squaring both sides

• Dividing both sides by a constant

• Adding a constant to both sides

In the equation \( x^2 - 9x + 14 = 0 \), what are the factors?

• (x - 1)(x - 14)

• (x - 2)(x - 7)

• (x - 4)(x - 5)

• (x - 3)(x - 6)

Which solution is extraneous for the equation \( \sqrt{x+2} + 4 = x \)?

• x = 4

• x = 5

• x = 7

• x = 2

What is the correct solution for the equation \( \sqrt{x+2} + 4 = x \)?

• x = 7

• x = 2

• x = 5

• x = 9

In the second example, what is the first step to solve \( \sqrt{2x} = x - 4 \)?

• Subtract 2x from both sides

• Multiply both sides by 2

• Add 4 to both sides

• Square both sides

What are the factors of the equation \( x^2 - 10x + 16 = 0 \)?

• (x - 5)(x - 5)

• (x - 4)(x - 6)

• (x - 2)(x - 8)

• (x - 3)(x - 7)

Which solution is extraneous for the equation \( \sqrt{2x} = x - 4 \)?

• x = 2

• x = 8

• x = 4

• x = 6

What is the correct solution for the equation \( \sqrt{2x} = x - 4 \)?

• x = 10

• x = 8

• x = 6

• x = 2

Why is it important to check for extraneous solutions in equations?

• To find the maximum value

• To ensure all solutions are valid

• To simplify the equation

• To determine the range of solutions