Answer :
To factor the expression
[tex]$$12w^2 - 147,$$[/tex]
we begin by identifying a common factor in both terms.
Step 1. Factor out the common factor
Notice that both coefficients, 12 and 147, are divisible by 3. We factor out 3:
[tex]$$12w^2 - 147 = 3 \left(\frac{12w^2}{3} - \frac{147}{3}\right) = 3(4w^2 - 49).$$[/tex]
Step 2. Factor the quadratic expression
Now, we focus on factoring the quadratic expression inside the parentheses, [tex]$4w^2 - 49$[/tex]. This is a difference of squares because:
- [tex]$4w^2$[/tex] can be written as [tex]$(2w)^2$[/tex], and
- [tex]$49$[/tex] can be written as [tex]$7^2$[/tex].
Recall that a difference of squares factors as:
[tex]$$a^2 - b^2 = (a - b)(a + b).$$[/tex]
Thus, letting [tex]$a=2w$[/tex] and [tex]$b=7$[/tex], we have:
[tex]$$4w^2 - 49 = (2w)^2 - 7^2 = (2w - 7)(2w + 7).$$[/tex]
Step 3. Write the final factored form
Substituting the factorized quadratic back, we get:
[tex]$$12w^2 - 147 = 3(2w - 7)(2w + 7).$$[/tex]
Thus, the expression in factored form is:
[tex]$$\boxed{3(2w - 7)(2w + 7)}.$$[/tex]
[tex]$$12w^2 - 147,$$[/tex]
we begin by identifying a common factor in both terms.
Step 1. Factor out the common factor
Notice that both coefficients, 12 and 147, are divisible by 3. We factor out 3:
[tex]$$12w^2 - 147 = 3 \left(\frac{12w^2}{3} - \frac{147}{3}\right) = 3(4w^2 - 49).$$[/tex]
Step 2. Factor the quadratic expression
Now, we focus on factoring the quadratic expression inside the parentheses, [tex]$4w^2 - 49$[/tex]. This is a difference of squares because:
- [tex]$4w^2$[/tex] can be written as [tex]$(2w)^2$[/tex], and
- [tex]$49$[/tex] can be written as [tex]$7^2$[/tex].
Recall that a difference of squares factors as:
[tex]$$a^2 - b^2 = (a - b)(a + b).$$[/tex]
Thus, letting [tex]$a=2w$[/tex] and [tex]$b=7$[/tex], we have:
[tex]$$4w^2 - 49 = (2w)^2 - 7^2 = (2w - 7)(2w + 7).$$[/tex]
Step 3. Write the final factored form
Substituting the factorized quadratic back, we get:
[tex]$$12w^2 - 147 = 3(2w - 7)(2w + 7).$$[/tex]
Thus, the expression in factored form is:
[tex]$$\boxed{3(2w - 7)(2w + 7)}.$$[/tex]
