Solving Radicals and Degenerate Cases with Equations

In this article, we will explore the methods and techniques for solving equations involving radicals. Specifically, we will focus on how to handle degenerate cases and extraneous solutions that may arise during the process. The primary example will illustrate the pitfalls of squaring both sides and the importance of verifying all potential solutions.

Introduction to Radical Equations

A radical equation is an equation that contains one or more radicals (square roots, cube roots, etc.). These equations can often be solved by isolating the radicals and then squaring or raising both sides to a power to eliminate the radicals. However, this process can sometimes introduce extraneous solutions, which must be checked to ensure they satisfy the original equation.

Example 1: Solving 3x 1^{1/2} - 1 x^{1/2}

Let's consider the equation:

3x 1^{1/2} - 1 x^{1/2}

Here are the steps to solve this equation:

Isolate the square root:

3x 1 - 1 x^{1/2}

Square both sides:

(3x 1 - 1)^2 (x^{1/2})^2

3x 1 x - 2x^{1/2} - 1

Rearrange the equation:

3x 1 - x 1 -2x^{1/2}

2x 2 -2x^{1/2}

x 1 -x^{1/2}

Square both sides again:

(x 1)^2 (-x^{1/2})^2

x^2 2x 1 x

x^2 2x 1 - x 0

x^2 x 1 0

Solve for x:

x(x 1) 0

x 0 or x -1

Check for extraneous solutions:

For x 0

3(0) 1^{1/2} - 1 0^{1/2}

0 1 - 1 0

0 0

0 0 is a true statement, but the original equation does not hold since 1^{1/2} - 1 ! 0

For x -1

3(-1) 1^{1/2} - 1 (-1)^{1/2}

-3 1 - 1 -1

-3 -1

-3 -1 is a false statement, hence x -1 is not a solution

Therefore, the original equation has no valid solutions.

General Approach for Radicals and Divergence

When dealing with more complex equations involving sums or differences of radicals, a general approach can be used to avoid extraneous solutions. Consider the equation:

sqrt(3x 1) - sqrt(x) -1

The approach is to use an identity to rewrite the equation in a more manageable form.

Using Identities for Simplification

Evaluate the left-hand side identity:

3x 1 - x 2x 1

Reformulate the identity in a more useful form:

(sqrt(3x 1))^2 - (sqrt(x))^2 2x 1

Divide by the original equation:

(sqrt(3x 1) - sqrt(x))(sqrt(3x 1) sqrt(x)) 2x 1

cfrac{(sqrt(3x 1))^2 - (sqrt(x))^2}{sqrt(3x 1) - sqrt(x)} cfrac{2x 1}{-1}

Factorize the numerator:

cfrac{sqrt(3x 1) - sqrt(x)sqrt(3x 1) - sqrt(x)}{sqrt(3x 1) - sqrt(x)} cfrac{2x 1}{-1}

cfrac{sqrt(3x 1) - sqrt(x)}{1} -2x - 1

Eliminate the radical:

sqrt(3x 1) - sqrt(x) -2x - 1

x 0

Substitute x 0 into the original equation:

sqrt(3(0) 1) - sqrt(0) -1

sqrt(1) - 0 -1

1 -1

This is false, hence x 0 is not a valid solution

Therefore, there is no valid solution for this equation.

Conclusion

When solving radical equations, it is crucial to be cautious about extraneous solutions introduced by squaring both sides. The approach of using identities and factoring can help simplify the equations and avoid false solutions. Always verify the solutions in the original equation to ensure they are valid.

References

This article is based on the principle of verifying solutions and using identities to avoid extraneous solutions in radical equations.