Problem 1 - Algebra A Combined Function

Question

Solve. $$ \sqrt{2 x}=4 $$

Step-by-Step Solution

Answer

The value of \( x \) is 8.

Step 1: Understanding the Equation

The equation given is \( \sqrt{2x} = 4 \). This means that the square root of \( 2x \) is equal to 4. Our task is to find the value of \( x \) that satisfies this equation.

Step 2: Isolate the Square Root

To solve for \( x \), we need to eliminate the square root. The next step is to square both sides of the equation to undo the square root: \( (\sqrt{2x})^2 = 4^2 \).

Step 3: Simplify Both Sides

Squaring the left side, we get \( 2x \), and squaring the right side, we get \( 16 \). So, the equation becomes \( 2x = 16 \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation by 2. This gives \( x = \frac{16}{2} \), which simplifies to \( x = 8 \).

Key Concepts

Solving Equations

Solving equations is like unraveling a mystery. We have an unknown value, often represented as \(x\), that we need to figure out. When we solve an equation, our goal is to find what number replaces \(x\) to make the equation true. In this problem, we have \( \sqrt{2x} = 4 \), where we must determine the value for \(x\) such that the expression is valid. The key to solving equations is to perform operations that bring us closer to isolating \(x\) on one side. This process often involves reversing operations, using techniques like algebraic manipulation and squaring both sides, to simplify and solve the equation step by step.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to transform equations for easier solving. In the context of this problem, algebraic manipulation helps us isolate \(x\). Here's how it happens:

• Identify terms that are grouped or under a radical, like the square root here.
• Apply mathematical operations to both sides of the equation to simplify these terms.
• Carefully perform each step to maintain the equation’s balance.

By manipulating the original equation \( \sqrt{2x} = 4 \), we aim to simplify it to \(2x = 16\), where the variable becomes clearer. Manipulation should preserve equality throughout, ensuring any transformation is applied equally to both sides.

Squaring Both Sides

Squaring both sides of an equation is a powerful technique to remove square roots. When faced with an equation like \(\sqrt{2x} = 4\), the goal is to eliminate the square root to simplify solving for \(x\). Here is how:

• Square the left side: \((\sqrt{2x})^2 = 2x\), effectively canceling out the square root.
• Square the right side: \(4^2 = 16\), to maintain balance in the equation.

Now, we have \(2x = 16\). This transformation is useful as it changes the form of the equation, making it a simple linear equation. Importantly, this step must be carried out carefully, as both sides need to be squared, ensuring the equation's integrity remains intact.