Is 2x√(1-x^2) √(4x^2-4x^4) True? Analyzing the Equation

Much confusion surrounds the equation 2x√(1-x^2) √(4x^2-4x^4). While it appears plausible, a deeper look into the algebraic and trigonometric underpinnings reveals the nuances that determine its validity. This article will delve into the details, providing a clear understanding of when and why this equation holds true.

Understanding the Domain of the Equation

The equation 2x√(1-x^2) √(4x^2-4x^4) is defined for values of x in the interval -1 ≤ x ≤ 1. This interval is crucial as it defines the domain within which the square roots are real numbers. Beyond this interval, the square roots might become imaginary or undefined.

Left-Hand Side Analysis: 2x√(1-x^2)

The left-hand side of the equation, 2x√(1-x^2), involves a product of a linear term x and the square root of a quadratic term (1-x^2). The term √(1-x^2) is only real within the interval -1 ≤ x ≤ 1. Moreover, for x in the interval (-1, 0) and x in (0, 1), the value of 2x√(1-x^2) can be positive or negative depending on the sign of x

Right-Hand Side Analysis: √(4x^2-4x^4)

The right-hand side of the equation, √(4x^2-4x^4), can be simplified first. By factoring out the common term, we get √(4x^2(1-x^2)) 2x√(1-x^2). This simplification is valid under the condition that 1-x^2 ≥ 0, which is true for -1 ≤ x ≤ 1.

However, the square root of a real number is always non-negative. Therefore, the right-hand side simplifies to 2|x|√(1-x^2). This means that the right-hand side will always be non-negative, regardless of the sign of x.

Comparing Left-Hand and Right-Hand Sides

The left-hand side, 2x√(1-x^2), can be negative, zero, or positive, depending on the value of x. The right-hand side, 2|x|√(1-x^2), is always non-negative. Therefore, for any value of x in the interval (-1, 0) or (0, 1), the left-hand side and the right-hand side will not be equal unless the sign of x is the same in both expressions.

Equality only holds when x is non-negative, specifically for the interval 0 ≤ x ≤ 1. This is because for x in this interval, the absolute value |x| is equal to x, and hence, the expressions are equivalent.

Conclusion and Key Takeaways

To summarize, the equation 2x√(1-x^2) √(4x^2-4x^4) is only true for values of x in the interval [0, 1]. For x in the interval [-1, 0), the equation does not hold. This is due to the absolute value in the simplified form of the right-hand side, which negates the possibility of negative values.

Understanding these nuances is crucial for correctly evaluating algebraic and trigonometric equations. Always check the domain and ensure that all terms are handled appropriately, especially when dealing with square roots and absolute values.

Keywords: equation verification, algebraic simplification, trigonometric identities