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Chapter 10: Problem 1

Why is it necessary to check the proposed solutions to a radical equation in the original equation?

Short Answer

Expert verified
It is necessary to check the proposed solutions to a radical equation in the original equation to differentiate between true solutions and extraneous solutions. Extraneous solutions are introduced during the solving process but do not actually satisfy the original equation. By checking the proposed solutions, we can eliminate such extraneous solutions and validate the true solutions that satisfy the given equation.

Step by step solution

01

Problem

Given a radical equation that you need to solve, the first thing that you have to do is to identify the true solutions and extraneous solutions.
02

Isolate the radical term

For any given radical equation, the first step would be to isolate the radical term on one side of the equation. For example, consider the following equation: \(\sqrt{x} = 2\). In this case, the radical term (the square root of x) is already isolated on the left-hand side of the equation.
03

Eliminate the radical

After isolating the radical term, the next step is to eliminate the radical by raising both sides of the equation to the power of the radical index. In the example given above (\(\sqrt{x} = 2\)), you would square both sides of the equation to get rid of the square root: \((\sqrt{x})^2=2^2\), which simplifies to \(x = 4\).
04

Solve the equation without the radical

Now that you have gotten rid of the radical, solve the remaining equation (in our example, \(x = 4\)) to find the proposed solution(s). In this case, there is only one solution: \(x = 4\).
05

Check proposed solution(s) in the original equation

Now that you have the proposed solution(s), check them in the original equation to eliminate any extraneous solutions. Plug the value of x back into the original equation and see if it works. In our example, substitute x=4 back to the original equation: \(\sqrt{4} = 2\). Since \(2 = 2\), it turns out that this solution satisfies the original equation. Thus, we've validated that \(x = 4\) is a true solution. In conclusion, it is necessary to check the proposed solutions to a radical equation in the original equation to ensure that we are not including extraneous solutions, which only arise during the solution process but do not satisfy the original equation.

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