Find the value of D = 9x² − 12xy + 4y² given that 3x − 2y = 20.

Step-by-step solution

1. 1

   Take the square roots of the two square terms. For the pattern $A^2 - 2AB + B^2$ you need $A$ and $B$ first:

   $A^2 = 9x^2 \Rightarrow A = 3x, \qquad B^2 = 4y^2 \Rightarrow B = 2y$

   Here the leading coefficients $9$ and $4$ are themselves perfect squares, which is the signal that this identity applies.

2. 2

   Verify the middle term before committing. The pattern requires the middle term to be exactly $2AB$:

   $2AB = 2(3x)(2y) = 12xy$

   It matches the $-12xy$ in the expression, with the minus sign selecting $(A - B)^2$ rather than $(A + B)^2$. If the middle coefficient had been anything other than $12$, the expression would not be a perfect square at all.

3. 3

   Rewrite as a single square.

   $D = 9x^2 - 12xy + 4y^2 = (3x - 2y)^2$

4. 4

   Substitute the given relation. With $3x - 2y = 20$:

   $D = 20^2 = 400$

   Again the whole bracket is replaced at once; there is not enough information to pin down $x$ and $y$ individually, and none is needed.

5. 5

   Check with two concrete solutions of 3x − 2y = 20. Take $y = 2$, $x = 8$: $9(64) - 12(16) + 4(4) = 576 - 192 + 16 = 400 \ \checkmark$. Take $y = -1$, $x = 6$: $324 + 72 + 4 = 400 \ \checkmark$. The value is independent of which solution is used, as expected.