If 2x - 3y=8 and xy =2 then find the value of 8x³ - 27y³?
- 780
- 832
- 800
- None of these
Explanation
We know the identity:
a³ - b³ = (a - b)(a² + ab + b²)
Here, 8x³ - 27y³ = (2x)³ - (3y)³
= (2x - 3y)((2x)² + (2x)(3y) + (3y)²)
Given 2x - 3y = 8 and xy = 2:
(2x)² + (2x)(3y) + (3y)² = 4x² + 6xy + 9y²
= 4x² + 6(2) + 9y²
= 4x² + 12 + 9y²
We need to express 4x² + 9y² in terms of 2x - 3y and xy:
(2x - 3y)² = 4x² - 12xy + 9y²
64 = 4x² - 12(2) + 9y²
64 = 4x² - 24 + 9y²
4x² + 9y² = 88
Now, 4x² + 6xy + 9y² = 88 + 12 = 100
So, 8x³ - 27y³ = (2x - 3y)((2x)² + (2x)(3y) + (3y)²)
= 8 × 100
= 800
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- (t² + 4)(t - 2)(t - 2)
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