Question

Find the real solution(s) of the radical equation. Check your solutions. \(\sqrt{2 x}-10=0\)

Short answer

The real solution of the radical equation \(\sqrt{2 x}-10=0\) is \(x = 50\).

Step-by-step solution

Isolate the square root

Firstly, isolate \(\sqrt{2 x}\) on one side of the equation. This can be done by adding 10 to both sides which gives \(\sqrt{2x} = 10\)

Remove the square root

To remove the square root, we square both sides of the equation. This gives \(2x = 100\) after simplification.

Isolate the variable x

Now, solve for x by dividing both sides by 2. This gives \(x = 50\).

Check the solution

To check the solution, substitute \(x = 50\) into the original equation. Once you do this, you get \(\sqrt{2 * 50}-10=0\) which simplifies to \(10-10=0\). This verifies that \(x = 50\) is a valid solution.

Key concepts

Solving Square Root Equations

Solving square root equations is a critical skill in algebra, especially as these equations often appear in various mathematical contexts. To solve such equations, one typically needs to isolate the square root expression and then eliminate the square root by squaring both sides of the equation.

Consider the equation \(\sqrt{2x} - 10 = 0\). The first step is to isolate the square root term, which yields \(\sqrt{2x} = 10\). Next, to remove the square root, we square both sides resulting in \(2x = 100\).

• Isolation of square root: \(\sqrt{2x} - 10 = 0 \to \sqrt{2x} = 10\)
• Squaring both sides: \(\left(\sqrt{2x}\right)^2 = 10^2\) leading to \(2x = 100\).

These steps transform the radical equation into a simpler linear equation that can be easily solved by further isolating the variable.

Isolating Variables

Isolating the variable is a foundational technique used to find the solution to algebraic equations. The goal here is to rearrange the equation so that the unknown variable stands alone on one side of the equation with a coefficient of one.

Following the removal of the square root in our previous example, we now have a linear equation, \(2x = 100\). To isolate \(x\), the next logical step is to divide both sides of the equation by 2. This gives us \(x = 50\).

• Division by the coefficient: \(2x = 100 \to x = \frac{100}{2}\)
• Final Solution: \(x = 50\)

By isolating the variable, the equation becomes straightforward, allowing for quick and clear identification of the solution.

Radical Equation Check

After obtaining a potential solution to a radical equation, it is essential to perform a 'radical equation check' to ensure its validity. The reason behind this check is to avoid extraneous solutions—answers that result from the algebraic steps but do not satisfy the original equation.

To check the solution \(x = 50\) in our given equation, we substitute it back into the original equation \(\sqrt{2x} - 10 = 0\). Following the substitution, we have \(\sqrt{2 \cdot 50} - 10 = 0\) which simplifies to \(10 - 10 = 0\), confirming the correctness of our solution.

• Substitute back: \(\sqrt{2 \cdot 50} - 10 \to 10 - 10\)
• Verification: The resulting \(0 = 0\) indicates that \(x = 50\) is indeed a valid solution.

This step is imperative to ensure the solution is legitimate in the context of the original problem.