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Simplify: 4x2−12xy+9y216x4−72x2y2+81y4−(2x−3y), given that 2x>3y.
4x2
6y
4x
6y2
6y
Observe the terms inside the square roots are perfect squares: (4x2−9y2)2 would not work, but (4x2−9y2) is a difference of squares; look at (4x2)2−2(4x2)(9y2)+(9y2)2.
Expression:\frac{\sqrt{16x⁴ - 72x²ʸ^2 + 81y⁴}}{\sqrt{4x² - 12xy + 9y²}} - (2x - 3y) with condition 2x > 3y.
(a−b)2=a2−2ab+b2, x2=∣x∣
Observe the terms inside the square roots are perfect squares: (4x2−9y2)2 would not work, but (4x2−9y2) is a difference of squares; look at (4x2)2−2(4x2)(9y2)+(9y2)2.
Ignoring the square root property A2=∣A∣. Since it is given 2x>3y, the expression 2x−3y is positive, but the numerator simplifies to ∣4x2−9y2∣.
Simplify Numerator
The numerator is 16x4−72x2y2+81y4. This is a perfect square trinomial of the form (4x2−9y2)2.
(4x2−9y2)2=∣4x2−9y2∣
Simplify Denominator
The denominator is 4x2−12xy+9y2. This is a perfect square trinomial (2x−3y)2.
(2x−3y)2=∣2x−3y∣
Evaluate with Given Condition
Given 2x>3y, the term (2x−3y) is positive. Also, 4x2−9y2=(2x−3y)(2x+3y). Since both x,y are typically positive in such problems, 4x2−9y2 is positive. Thus, the expression becomes 2x−3y(2x−3y)(2x+3y)−(2x−3y)
(2x+3y)−(2x−3y)
Final Result
Subtract the terms: 2x+3y−2x+3y=6y.
6y
B is correct because the expression simplifies to (2x+3y)−(2x−3y)=6y.
Recognizing perfect squares a2±2ab+b2 is essential for both competitive algebra and geometry coordinate problems.
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