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Answer
\( 4x^{2} - y^{4} = (2x - y^{2})(2x + y^{2}) \)
Solution
We can recognize that the expression 4x² - y⁴ is a difference of two squares.
Step 1: Write 4x² and y⁴ as squares.
4x² = (2x)² and y⁴ = (y²)²
Step 2: Apply the difference of squares formula:
a² - b² = (a - b)(a + b)
Here, a = 2x and b = y².
Thus, the factorised form is:
(2x - y²)(2x + y²)
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Simplify this solution Bonus Knowledge
To factor \( 4x^{2} - y^{4} \), you can recognize it as a difference of squares. It can be rewritten as \( (2x)^{2} - (y^{2})^{2} \). Using the difference of squares formula \( a^{2} - b^{2} = (a - b)(a + b) \), we have: \[ (2x - y^{2})(2x + y^{2}). \] Now, we’ve successfully expressed \( 4x^{2} - y^{4} \) in its factored form! Another fun way to look at this factorization is by considering the geometric interpretation. The expression \( 4x^{2} - y^{4} \) represents a hyperbola in the xy-plane. The factored form helps identify the curves and their intersections more clearly, adding an exciting layer to the algebra!
