To determine which value of [tex]\( a \)[/tex] allows the expression [tex]\( x^2 - a \)[/tex] to be completely factored, we need to recognize the form of the expression and how it can be factored.
The expression [tex]\( x^2 - a \)[/tex] is a quadratic expression. For it to be completely factored, [tex]\( a \)[/tex] should be a perfect square. When it is a perfect square, we can write the expression as a difference of squares:
[tex]\[ x^2 - a = (x - \sqrt{a})(x + \sqrt{a}) \][/tex]
Let's examine each given value of [tex]\( a \)[/tex] to see if it is a perfect square:
1. For [tex]\( a = 12 \)[/tex]:
[tex]\[ \sqrt{12} \][/tex] is not an integer. Therefore, [tex]\( x^2 - 12 \)[/tex] does not factor into a product of binomials with integer coefficients.
2. For [tex]\( a = 36 \)[/tex]:
[tex]\[ \sqrt{36} = 6 \][/tex]
Thus, [tex]\( x^2 - 36 \)[/tex] can be factored as:
[tex]\[ x^2 - 36 = (x - 6)(x + 6) \][/tex]
It is completely factored.
3. For [tex]\( a = 49 \)[/tex]:
[tex]\[ \sqrt{49} = 7 \][/tex]
Thus, [tex]\( x^2 - 49 \)[/tex] can be factored as:
[tex]\[ x^2 - 49 = (x - 7)(x + 7) \][/tex]
It is completely factored.
4. For [tex]\( a = 81 \)[/tex]:
[tex]\[ \sqrt{81} = 9 \][/tex]
Thus, [tex]\( x^2 - 81 \)[/tex] can be factored as:
[tex]\[ x^2 - 81 = (x - 9)(x + 9) \][/tex]
It is completely factored.
From our analysis, the values [tex]\( a = 36 \)[/tex], [tex]\( a = 49 \)[/tex], and [tex]\( a = 81 \)[/tex] all allow the expression [tex]\( x^2 - a \)[/tex] to be completely factored. However, [tex]\( a = 12 \)[/tex] does not.
Therefore, any of the values [tex]\( 36 \)[/tex], [tex]\( 49 \)[/tex], or [tex]\( 81 \)[/tex] are correct answers as they make the expression [tex]\( x^2 - a \)[/tex] completely factored.
