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If 3x+2y=12 and xy=6, find the value of 9x2+4y2.
72
84
96
108
72
Notice the identity (3x+2y)2=(3x)2+(2y)2+2(3x)(2y). Since 3x+2y=12 and xy=6, the result is 122−12(xy)=144−72=72.
Two equations are provided: 3x + 2y = 12 and xy = 6.
(a+b)2=a2+2ab+b2
Notice the identity (3x+2y)2=(3x)2+(2y)2+2(3x)(2y). Since 3x+2y=12 and xy=6, the result is 122−12(xy)=144−72=72.
Students often calculate (3x)2+(2y)2 as 3x2+2y2 or 9x2+4y2+2xy instead of 9x2+4y2+12xy.
Square the first equation
To relate the linear equation to the quadratic terms 9x2 and 4y2, square the equation 3x+2y=12.
(3x+2y)2=122
Expand using algebraic identity
Apply the identity (a+b)2=a2+2ab+b2 where a=3x and b=2y.
(3x)2+(2y)2+2(3x)(2y)=144
Substitute the known value of xy
Simplify the expansion and substitute xy=6 into the middle term 12xy.
9x2+4y2+12(6)=144
Solve for 9x² + 4y²
Perform the subtraction 144−72 to find the final value.
9x2+4y2=144−72=72
A is correct because the expansion of (3x+2y)2 leads to 9x2+4y2+72=144, which simplifies to 72.
This identity-based approach is frequently tested in quadratic equations and polynomial factorization problems across various competitive exams.
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