Given the quadratic equation [ x² − (A − 3)x − (A − 2) = 0, ] for what value of (A) will the difference of the squares of the roots be zero?

To find the value of $A$ for which the difference of the squares of the roots of the quadratic equation $x^2 - (A - 3)x - (A - 2) = 0$ is zero, we start by noting that the difference of the squares of the roots $r_1$ and $r_2$ can be expressed as:

$$r_1^2 - r_2^2 = (r_1 - r_2)(r_1 + r_2)$$

For this difference to be zero, either $r_1 - r_2 = 0$ (which means the roots are equal) or $r_1 + r_2 = 0$.

Using Vieta's formulas, we know:

• The sum of the roots $r_1 + r_2 = A - 3$
• The product of the roots $r_1 r_2 = -(A - 2)$

For the roots to be equal, the discriminant must be zero:

$$D = b^2 - 4ac = (A - 3)^2 - 4(1)(-(A - 2)) = 0$$

Expanding this gives:

$$(A - 3)^2 + 4(A - 2) = 0$$

Expanding further:

$$A^2 - 6A + 9 + 4A - 8 = 0$$

Combining like terms:

$$A^2 - 2A + 1 = 0$$

Factoring:

$$(A - 1)^2 = 0$$

Thus, $A - 1 = 0$ implies $A = 1$.

Therefore, the value of $A$ for which the difference of the squares of the roots is zero is:

$$\boxed{1}$$