Question

Compute the value of 9x2 + 4y2 if xy = 6 and 3x + 2y = 12.​

Answer 1

Given:

• Two equations, 3x + 2y = 12 and x*y =6.

To Find:

• The value of 9x²+4y² using the above equations.

Solution:

The given problem can be solved using the expansion of (a+b)².

1. The given equations are 3x + 2y = 12 and x*y =6,
2. Consider the equation 3x 2y = 12

✿ Apply sqaure on both the sides of the equation,

⮕ (3x + 2y) ^ 2 = (12) ^ 2

⮕ 9x ^ 2 + 4y ^ 2 + 12xy = 144 ( Using the expansion (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2ab )

⮕ 9x ^ 2 + 4y ^ 2 = 144 - 12xy

⮕ 9x ^ 2 + 4y ^ 2 = 144 - 12(6) (As y = 6) ,

⮕ 9x ^ 2 + 4y ^ 2 = 144 - 72 ,

⮕ 9x ^ 2 + 4y ^ 2 = 72 .

Therefore, the value of 9x² + 4y² is 72.

Answer 2

72

Explanation:

We can try square 3x + 2y = 12 on both sides. This will result in:

`\displaystyle{\left(3x+2y\right)^2=12^2}\\\\\displaystyle{9x^2+12xy+4y^2=144}`

Since the value of xy is equal to 6. Thus, substitute xy = 6:

`\displaystyle{9x^2+12\cdot 6+4y^2=144}\\\\\displaystyle{9x^2+72+4y^2=144}`

Subtract 72 on both sides to solve for 9x² + 4y². Thus:

`\displaystyle{9x^2+4y^2=144-72}\\\\\displaystyle{9x^2+4y^2=72}`

Another method is that we apply the perfect square on:

`\displaystyle{\left(3x+2y\right)^2 = 9x^2+12xy+4y^2}`

But since we only want to compute 9x² + 4y², we subtract 12xy away. This will leave us with:

`\displaystyle{9x^2+4y^2=\left(3x-2y\right)^2-12xy}`

Since 3x - 2y = 12 and xy = 6 then we substitute in, we have:

`\displaystyle{9x^2+4y^2=12^2-12(6)}\\\\\displaystyle{9x^2+4y^2=144-72}\\\\\displaystyle{9x^2+4y^2 = 72}`