Question

\(2 \sqrt{x-4}-2=2\)

Short answer

The solution to the equation is \(x=8\)

Step-by-step solution

Isolate the square root

Add 2 to both sides of equation: \(2 \sqrt{x-4}=4\)

Step 2:Keep sqrt term alone

Divide both sides of the equation by 2 to get the square root term alone: \(\sqrt{x-4}=2\)

Square both sides of the equation

Square both sides of the equation to eliminate the square root: \((\sqrt{x-4})^2=2^2\). This simplifies to \(x-4=4\)

Solve for x

Add 4 to both sides of the equation to solve for 'x': \(x=8\)

Key concepts

Isolating Square Roots

Isolating square roots is often the first and key step when solving equations involving square root terms. This process involves getting the square root by itself on one side of the equation. You want to do this to simplify the problem and prepare it for the next step, which usually involves eliminating the root.
To isolate the square root, you may need to move other terms to the opposite side of the equation. In the given exercise, we started with the equation:
\[2 \sqrt{x-4} - 2 = 2\]
By adding 2 to both sides, we make progress towards isolating the square root:
\[2 \sqrt{x-4} = 4\]

• Identify terms that need to be moved.
• Add or subtract terms to both sides to eliminate these terms from the square root side.

Once you've removed the surrounding elements, the square root stands alone, ready for further manipulation.

Squaring Both Sides

After you've successfully isolated the square root, the next step is typically to eliminate it by squaring both sides of the equation. Squaring is useful because it reverses the square root operation, thus allowing you to work with a more straightforward algebraic expression.
Here, the isolated square root equation is:
\[\sqrt{x-4} = 2\]
When you square both sides, you must remember that this operation applies to both the square root and the constant:
\[(\sqrt{x-4})^2 = 2^2\]
This transforms the equation into:
\[x-4 = 4\]

• Ensure both sides are squared; don't forget any terms.
• Use squaring judiciously to avoid errors.

By squaring both sides, the square root is effectively 'removed,' leading to simpler equations to solve.

Algebraic Manipulation

Algebraic manipulation involves reorganizing and simplifying equations to find the value of the variable. This term covers various operations, but in this particular case, it played a role after squaring both sides of the equation.
Once you have the equation:
\[x-4 = 4\]
your goal is to solve for the unknown variable. To do this, you need to perform operations that will both simplify the equation and isolate the unknown:

• Add 4 to both sides to undo the subtraction.
• Ensure all operations maintain the equation's balance.

This gives:
\[x = 8\]
Algebraic manipulation helps in solving equations by systematically working through, rearranging and simplifying expressions to find what you're looking for.

Linear Equations

Once you've eliminated the square root, oftentimes you'll end up with a linear equation, which is simpler to solve. A linear equation is typically in the form of \(ax + b = c\). In the exercise, after simplifying, you've transformed a square root equation into a linear one:
\[x-4 = 4\]
Linear equations are straightforward because they involve variables to the first power and can often be solved in just a few steps. In your case, solving means finding the value of \(x\):

• Rearrange the equation to get all variable terms on one side.
• Move constants to the other side using addition or subtraction.

This led you to the solution where \(x = 8\). Linear equations highlight the simplicity of the final step in finding the unknown in comparison to tasks like isolating square roots or squaring terms.
Understanding how to solve these forms the base for dealing with more complex equations efficiently.