Sure, let's factor the trinomial [tex]\(16d^2 - 24d + 9\)[/tex].
### Step-by-Step Solution
1. Identify the coefficients:
- The coefficient of [tex]\(d^2\)[/tex] is [tex]\(16\)[/tex].
- The coefficient of [tex]\(d\)[/tex] is [tex]\(-24\)[/tex].
- The constant term is [tex]\(9\)[/tex].
2. Consider the form of a quadratic trinomial:
A quadratic trinomial in the form [tex]\(ax^2 + bx + c\)[/tex] can often be written as the product of two binomials. For this trinomial, we will rewrite it in the form [tex]\((md + n)^2\)[/tex], where [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are constants.
3. Recognize that the trinomial is a perfect square:
Observing the given trinomial [tex]\(16d^2 - 24d + 9\)[/tex], we notice it resembles the format of a perfect square trinomial. Generally, a perfect square trinomial is in the form [tex]\((md + n)^2 = (md + n)(md + n)\)[/tex].
4. Finding [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
- Since [tex]\(16d^2\)[/tex] is the leading term, [tex]\(m\)[/tex] must be [tex]\(4\)[/tex] because [tex]\(4^2 = 16\)[/tex].
- Similarly, the constant term [tex]\(9\)[/tex] suggests that [tex]\(n\)[/tex] must be [tex]\(3\)[/tex] because [tex]\(3^2 = 9\)[/tex].
- Consider the middle term, [tex]\(-24d\)[/tex]. For a perfect square trinomial [tex]\((md + n)^2\)[/tex], the middle term should be [tex]\(2 \cdot m \cdot n \cdot d\)[/tex]. Substitute [tex]\(m = 4\)[/tex] and [tex]\(n = 3\)[/tex]:
[tex]\[
2 \cdot 4 \cdot 3 = 24
\][/tex]
- The middle term is [tex]\(-24d\)[/tex], which matches the provided trinomial when considered with negative signs.
5. Write the perfect square trinomial:
Given the coefficients match for [tex]\(16d^2\)[/tex], [tex]\(-24d\)[/tex], and [tex]\(9\)[/tex], we can conclude:
[tex]\[
16d^2 - 24d + 9 = (4d - 3)^2
\][/tex]
### Conclusion
The factorization of the trinomial [tex]\(16d^2 - 24d + 9\)[/tex] is:
[tex]\[
(4d - 3)^2
\][/tex]
This completes the factorization of the given quadratic trinomial.
