Answer :
To simplify the expression
[tex]$$
3x^2 - 27,
$$[/tex]
follow these steps:
1. Factor out the common factor:
Both terms in the expression have a common factor of 3. Factoring out 3 gives:
[tex]$$
3x^2 - 27 = 3(x^2 - 9).
$$[/tex]
2. Recognize the difference of squares:
The expression inside the parentheses, [tex]$x^2 - 9$[/tex], is a difference of squares because it can be written as:
[tex]$$
x^2 - 9 = x^2 - 3^2.
$$[/tex]
The difference of squares formula states that:
[tex]$$
a^2 - b^2 = (a - b)(a + b),
$$[/tex]
where here [tex]$a = x$[/tex] and [tex]$b = 3$[/tex]. Thus, we can factor [tex]$x^2 - 9$[/tex] as:
[tex]$$
x^2 - 9 = (x - 3)(x + 3).
$$[/tex]
3. Write the fully factored form:
Substitute the factors back into the expression:
[tex]$$
3(x^2 - 9) = 3(x - 3)(x + 3).
$$[/tex]
Thus, the simplified form of the expression is:
[tex]$$
3(x-3)(x+3).
$$[/tex]
[tex]$$
3x^2 - 27,
$$[/tex]
follow these steps:
1. Factor out the common factor:
Both terms in the expression have a common factor of 3. Factoring out 3 gives:
[tex]$$
3x^2 - 27 = 3(x^2 - 9).
$$[/tex]
2. Recognize the difference of squares:
The expression inside the parentheses, [tex]$x^2 - 9$[/tex], is a difference of squares because it can be written as:
[tex]$$
x^2 - 9 = x^2 - 3^2.
$$[/tex]
The difference of squares formula states that:
[tex]$$
a^2 - b^2 = (a - b)(a + b),
$$[/tex]
where here [tex]$a = x$[/tex] and [tex]$b = 3$[/tex]. Thus, we can factor [tex]$x^2 - 9$[/tex] as:
[tex]$$
x^2 - 9 = (x - 3)(x + 3).
$$[/tex]
3. Write the fully factored form:
Substitute the factors back into the expression:
[tex]$$
3(x^2 - 9) = 3(x - 3)(x + 3).
$$[/tex]
Thus, the simplified form of the expression is:
[tex]$$
3(x-3)(x+3).
$$[/tex]
