Question

Solve the radical equation. Check all solutions to eliminate extraneous solutions and do not include them in your answer. If your answer is not an integer, then type it as a decimal rounded to the nearest hundredth. √(2x³) - √(x²) = 0. Find the value of x.

Answer 1

The equation √(2x³) - √(x²) = 0 has no solution, as the potential answer x = 0.50 is extraneous when checked against the original equation.

Step-by-step explanation:

To solve the radical equation √(2x³) - √(x²) = 0, we first isolate one of the radicals:

√(2x³) = √(x²)

Squaring both sides to eliminate the radicals gives us:

2x³ = x²

Dividing both sides by x², assuming x is not equal to 0, simplifies to:

2x = 1

So the solution for x is:

x = ½ or 0.50 when rounded to the nearest hundredth.

We need to check that this solution is not extraneous. Substituting x back into the original equation:

√(2(0.50)³) - √(0.50²) = √(0.50) - √(0.25)

0.71 - 0.50 = 0.21 ≀ 0

Since the check does not result in a true statement, the solution x = 0.50 is extraneous and there is no solution to the equation.