Given the statement "If $x^2 = 4$, then $x = 2$", we need to find the converse, inverse, and contrapositive statements. We must also determine the truth or falsity of the original statement and the three derived statements.

1. Problem Description

Given the statement "If $x^2 = 4$, then $x = 2$", we need to find the converse, inverse, and contrapositive statements. We must also determine the truth or falsity of the original statement and the three derived statements.

2. Solution Steps

Original Statement: If $x^2 = 4$, then $x = 2$.

If $x^2 = 4$, then $x$ can be either 2 or -

2. $x^2 = 4 \implies x = \pm 2$.

Therefore, the original statement is false.

Converse: If $x = 2$, then $x^2 = 4$.

If $x = 2$, then $x^2 = 2^2 = 4$.

Therefore, the converse statement is true.

Inverse: If $x^2 \ne 4$, then $x \ne 2$.

If $x^2 \ne 4$, then $x$ can be any real number except 2 and -

2. This implies $x$ can be something like 1, so $x\ne 2$.

Therefore, the inverse statement is false.

Contrapositive: If $x \ne 2$, then $x^2 \ne 4$.

If $x \ne 2$, then $x$ can take infinitely many values. If $x=-2$, then $x \ne 2$ but $x^2=(-2)^2 = 4$. Hence $x^2$ can equal

4. Therefore, the contrapositive statement is false.

3. Final Answer

Original statement: False

Converse: True

Inverse: False

Contrapositive: False