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Question
If 3x + 4y = 16 and xy = 4, find the value of 9x2 + 16y2.
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Solution
Given that (3x + 4y) = 16 and xy = 4,
We need to find 9x2 + 16y2,
We know that,
(a + b)2 = a2 + b2 + 2ab
Consider the square of 3x + 4y:
∴ (3x + 4y)2 = (3x)2 + (4y)2 + 2 × 3x × 4y
⇒ (3x + 4y)2 = 9x2 + 16y2 + 24xy ...(1)
Substitute the values of (3x + 4y) and xy,
In the above equation (1), we have,
(3x + 4y)2 = 9x2 + 16y2 + 24xy
⇒ (16)2 = 9x2 + 16y2 + 24(4)
⇒ 256 = 9x2 + 16y2 + 96
⇒ 9x2 + 16y2 = 256 − 96
∴ 9x2 + 16y2 = 160
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