Question

Solve for \(x\) and check. $$\sqrt{x}=6$$

Short answer

Solving \(\sqrt{x} = 6\) by squaring both sides gives \(x = 36\), which checks out when substituted back into the original equation.

Step-by-step solution

Square both sides

To eliminate the square root and solve for x, we need to square both sides of the equation. So, we will raise both sides of the equation to the power of 2.

Simplify the equation

After squaring both sides, we get: \(x = 6^2\). Now simplify the right side of the equation by calculating the square of 6.

Calculate the square of 6

Calculate the square of 6, which is \(6 \times 6 = 36\). So, the equation simplifies to \(x = 36\).

Check the solution

To verify the solution, substitute \(x = 36\) back into the original equation: \(\sqrt{36} = 6\). Since \(\sqrt{36}\) is indeed 6, the solution checks out.

Key concepts

Square Both Sides

When we encounter a square root equation, such as \(\sqrt{x} = 6\), one of our primary goals is to isolate the variable to solve for it. To achieve this, an essential strategy is to 'square both sides' of the equation.

Why do we square both sides? The reason behind this approach is that squaring is the inverse operation of taking the square root, which effectively cancels out the square root on one side and leaves us with the variable alone.

To execute this correctly, you must apply the squaring operation uniformly to both sides like so: \[ (\sqrt{x})^2 = 6^2 \.\] This action ensures that you are maintaining the balance of the equation, a fundamental principle in algebra. After squaring, you get \( x = 36 \), having removed the square root and made the equation easier to work with.

Simplify Equations

Following the squaring of both sides, simplifying equations is the subsequent step that requires precision and attentiveness. In our example, the squared equation \( x = 6^2 \) may seem straightforward, but every equation isn't always so simple.

As you continue to delve into more complex square root equations, you will encounter expressions that require the distributive property, combining like terms, and further simplification techniques after the squaring process. The key to mastering this stage is ensuring that each transformation adheres to algebraic rules and results in the most basic form of the equation, which usually means solving for \( x \).

Simplification may seem merely mechanical, but it's vital for recognizing the core structure of the equation and ensuring that all steps following it are based on a correct representation of the mathematical relationship.

Check Mathematical Solutions

The final step in solving square root equations - 'check mathematical solutions' - is sometimes neglected, but it is as crucial as solving the equation itself. After deducing that \( x = 36 \) is the solution to \(\sqrt{x} = 6\), validating this answer by plugging it back into the original equation is essential.

By substituting \( x \) with 36, \(\sqrt{36}\) should equal 6, which it does, as 36 is the perfect square of 6. For completeness, include a verification step whenever you solve equations: \[ \sqrt{x} \overset{?}{=} 6 \quad \text{becomes} \quad \sqrt{36} = \sqrt{6^2} = 6, \] thereby confirming the solution's accuracy.

Always remember, checking ensures that you haven't made any mistakes along the way, especially when squaring both sides, as it might introduce extraneous solutions. It's a necessary practice to confirm that the solution is not only mathematically sound but also meaningful in the context of the problem.