How to Solve Equations Involving Square Roots and Variables

When we encounter an equation such as $x^2 - 10cdotsqrt{x^3} 0$, it can be challenging to identify the correct solution method. This guide will walk you through the process step-by-step, providing a thorough understanding of how to approach such equations.

The Importance of the Domain

The solution to such an equation depends on the set from which $x$ is drawn. If $x$ is an integral domain, each factor can be set to zero and the solution can be found from the factors. However, if $x$ is not from an integral domain, you must specify the underlying domain of $x$. This is a critical step as different domains can lead to different solutions.

Solving the Equation Step-by-Step

Let's dissect the given equation: $x^2 - 10cdotsqrt{x^3} 0$.

Step 1: Rewrite the Equation

First, let's rewrite the equation by squaring both sides to remove the square root:

$x^2 - 10cdot(3x)^{1/3} 0$

Squaring both sides again to remove the square root:

$(x^2 - 10)^2 (3x)^{2/3}$

Step 2: Set the Factors to Zero

From the original equation, we can set the factors to zero:

$x^2 - 10 0$

$3x 0$

Solution 1: Setting $x^2 - 10 0$

Setting $x^2 - 10 0$ gives:

$x^2 10$

Taking the square root of both sides:

$x sqrt{10}$

or

$x -sqrt{10}$

Solution 2: Setting $3x 0$

Setting $3x 0$ gives:

$x 0$

or

$x -3$

Conclusion

Combining the solutions from both sets of equations, we get:

$x sqrt{10}$

$x -sqrt{10}$

$x 0$

$x -3$

It's important to check each solution in the original equation to ensure it is valid within the specified domain. This process of checking solutions is crucial to avoid extraneous solutions.

Key Takeaways

When solving equations involving square roots, consider the domain of the variable.

Square both sides to simplify the equation and then set each factor to zero.

Verify each solution in the original equation to ensure its validity.

Related Resources

To deepen your understanding of solving equations with square roots, consider exploring the following resources:

A collection of solved problems and exercises on algebraic equations with square roots.

Interactive tools and quizzes for practicing algebraic problem-solving skills.

Videos and tutorials on advanced algebraic concepts.