To factor the quadratic expression [tex]\( w^2 - 18w + 81 \)[/tex], we should recognize it as a perfect square trinomial. Let's go through the steps to identify and factor the expression correctly:
1. Identify the quadratic form: Our given expression is in the form [tex]\( w^2 - 18w + 81 \)[/tex], where the general form of a quadratic expression is [tex]\( ax^2 + bx + c \)[/tex].
2. Identify coefficients: Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = 81 \)[/tex].
3. Check for perfect square trinomial:
- A perfect square trinomial is of the form [tex]\( (x - k)^2 \)[/tex] or [tex]\( (x + k)^2 \)[/tex].
- It expands to [tex]\( x^2 \pm 2kx + k^2 \)[/tex].
4. Find values to match the given expression:
- For our [tex]\( w^2 - 18w + 81 \)[/tex] to be a perfect square, we need to match [tex]\( (w - k)^2 = w^2 - 2kw + k^2 \)[/tex].
- Comparing terms, [tex]\( -18w \)[/tex] should equal [tex]\( -2kw \)[/tex], which gives [tex]\( -2k = -18 \)[/tex]. Solving for [tex]\( k \)[/tex], we get [tex]\( k = 9 \)[/tex].
- Similarly, [tex]\( k^2 \)[/tex] should equal the constant term, [tex]\( 81 \)[/tex]. Substituting [tex]\( k = 9 \)[/tex], we see [tex]\( 9^2 = 81 \)[/tex].
5. Write it as a squared term:
- Given that [tex]\( k = 9 \)[/tex], the trinomial [tex]\( w^2 - 18w + 81 \)[/tex] can be factored as [tex]\( (w - 9)^2 \)[/tex].
So the factored form of [tex]\( w^2 - 18w + 81 \)[/tex] is:
[tex]\[ (w - 9)^2 \][/tex]
Thus, the final factored expression is:
[tex]\[ (w - 9)^2 \][/tex]
