Solving Radical and Polynomial Equations

This video tutorial focuses on solving radical equations, starting with basic examples and progressing to more complex ones. It covers techniques such as squaring both sides, handling cube roots, and checking for extraneous solutions. The tutorial also delves into factoring complex equations and emphasizes the importance of verifying solutions.

What is the first step in solving the equation \( \sqrt{3x + 1} = 4 \)?

• Subtract 1 from both sides

• Add 1 to both sides

• Square both sides

• Divide both sides by 3

In the equation \( \sqrt{7 - x} + 3 = 5 \), what should you do before squaring both sides?

• Add 3 to both sides

• Divide both sides by 2

• Subtract 3 from both sides

• Multiply both sides by 2

How do you solve the equation \( \sqrt[3]{x + 15} = 3 \)?

• Square both sides

• Add 15 to both sides

• Raise both sides to the third power

• Subtract 15 from both sides

What is the solution to the equation \( 2\sqrt{x} = x \)?

• x = 0 and x = 2

• x = 2 and x = 4

• x = 0 and x = 4

• x = 1 and x = 4

What is the first step in solving \( x^{1/4} + 4 = 7 \)?

• Divide both sides by 4

• Subtract 4 from both sides

• Multiply both sides by 4

• Add 4 to both sides

In the equation \( \sqrt{3x + 4} = \sqrt{4x + 3} \), what is the value of x?

• x = 0

• x = 1

• x = 2

• x = 3

When solving an equation with two radicals, what should you do first?

• Square both sides immediately

• Move one radical to the other side

• Subtract the radicals

• Add the radicals together

What is the correct factorization of the trinomial \( 9x^2 - 16x - 80 \)?

• (9x - 20)(x + 4)

• (9x + 4)(x - 20)

• (9x + 20)(x - 4)

• (3x - 4)(3x + 20)

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

• They are not necessary

• They can sometimes be incorrect

• They simplify the equation

• They always provide the correct answer

What is the extraneous solution in the equation \( \sqrt{5 + x} + 2 = \sqrt{4x + 9} \)?

• x = 4

• x = -20/9

• x = 5

• x = 0