A^2 + b^2 = c^2 + d^2 = 5, ac = bd = 3. Tính A = ad - bc

To find the value of A, we need to calculate ad - bc.

Given that ac = bd = 3, we can rewrite the equation a^2 + b^2 = c^2 + d^2 = 5 as (a^2 + b^2) - (c^2 + d^2) = 0.

Expanding this equation, we get a^2 + b^2 - c^2 - d^2 = 0.

Rearranging the terms, we have (a^2 - c^2) + (b^2 - d^2) = 0.

Using the difference of squares formula, we can factor this equation as (a + c)(a - c) + (b + d)(b - d) = 0.

Since ac = bd = 3, we can substitute these values into the equation:

(3)(a - c) + (3)(b - d) = 0.

Simplifying, we get 3a - 3c + 3b - 3d = 0.

Dividing both sides by 3, we have a - c + b - d = 0.

Rearranging the terms, we get a + b = c + d.

Now, let's solve for A = ad - bc.

We can rewrite A as A = (a + b)(d - c).

Using the equation a + b = c + d, we can substitute this into the expression for A:

A = (c + d)(d - c).

Expanding this expression, we get A = cd - c^2 + d^2 - cd.

Since ac = bd = 3, we can substitute these values into the equation:

A = 3 - c^2 + d^2 - 3.

Simplifying, we have A = d^2 - c^2.

Given that a^2 + b^2 = c^2 + d^2 = 5, we can substitute this into the equation:

A = 5 - 5.

Therefore, A = 0.