https://r2.erweima.ai/mathbot/image/1725574770708elq00hx1.jpg

Can you tells me how to get from the red to the blue?

To go from the red equation \( y = x \sqrt{2} \) to the blue equation \( y = \sqrt{\frac{x^2}{2}} \), we can follow these steps:

### Red Equation:
\[ y = x \sqrt{2} \]

### Step 1: Square both sides
To eliminate the square root in the red equation, we can square both sides:

\[
y^2 = (x \sqrt{2})^2
\]

### Step 2: Simplify the right-hand side
When squaring the right-hand side, apply the rule \( (a \cdot b)^2 = a^2 \cdot b^2 \):

\[
y^2 = x^2 \cdot (\sqrt{2})^2
\]
Since \( (\sqrt{2})^2 = 2 \), this becomes:
\[
y^2 = x^2 \cdot 2 = 2x^2
\]

### Step 3: Solve for \( y \)
Now, divide both sides by 2 to isolate \( x^2 \):

\[
\frac{y^2}{2} = x^2
\]

### Step 4: Express \( y \) as a square root
Finally, solve for \( y \) by taking the square root of both sides:

\[
y = \sqrt{\frac{x^2}{2}}
\]

This results in the blue equation.

---

Would you like further details on any of these steps, or have additional questions? Here are some related questions you might explore:

1. How does squaring both sides help in simplifying equations with square roots?
2. What happens if \( y = -x\sqrt{2} \), how does it affect the result?
3. Can the square root of a fraction like \( \frac{x^2}{2} \) be simplified further?
4. Why do we square both sides instead of directly manipulating the square root?
5. How does this method apply to more complex square root equations?

**Tip:** Always check the domain of your variable when working with square roots, since they only accept non-negative numbers unless you're dealing with complex numbers.

Can you tell me how to get from the red to the blue?

Understand how to simplify square root equations, going from y = x√2 to y = √(x² / 2). Follow a step-by-step guide explaining algebraic concepts like squaring both sides and simplifying square roots. Suitable for high school students.

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