To simplify the quadratic expression [tex]\(16d^2 - 24d + 9\)[/tex], let's consider any possible factoring or reduction:
Given:
[tex]\[16d^2 - 24d + 9\][/tex]
First, we can check if this quadratic expression can be factored. We look for two binomials of the form [tex]\((ad + b)(cd + e)\)[/tex] that when expanded will give us the original quadratic expression.
To factorize quadratic expressions, we look for a and b such that:
[tex]\[ac = 16 \quad \text{and} \quad be = 9\][/tex]
[tex]\[ad \cdot cd \rightarrow ad + cd = -24\][/tex]
After trying possible pairs, we find:
[tex]\[(4d - 3)(4d - 3) = 16d^2 - 12d - 12d + 9\][/tex]
Expanding each term:
[tex]\[16d^2 - 12d - 12d + 9 = 16d^2 - 24d + 9\][/tex]
So, the quadratic expression:
[tex]\[16d^2 - 24d + 9\][/tex]
can be written in its factored form as:
[tex]\[(4d - 3)^2\][/tex]
Therefore, the simplified form of the quadratic expression is:
[tex]\[ (4d - 3)^2 \][/tex]
