Which value of a would make the following expression completely factored x^2 – a?

Step 1: Define the Problem

Identify the value of $a$ that makes the expression $x^2 - a$ completely factored.

Step 2: Break Down the Problem

The expression $x^2 - a$ is a difference of squares, which can be factored as $(x + \sqrt{a})(x - \sqrt{a})$.

Step 3: Apply Relevant Concepts

For the expression to be completely factored, $a$ must be a perfect square. This is because the square root of $a$ must be a rational number for the expression to be factored into linear terms with rational coefficients.

Step 4: Analysis, Verify and Summarize

• If $a$ is a perfect square, say $a = b^2$, then the expression becomes $x^2 - b^2$.
• This can be factored as $(x + b)(x - b)$.

Final Answer

The value of $a$ that makes the expression $x^2 - a$ completely factored is any perfect square, $a = b^2$, where $b$ is an integer.