Factor completely.
$2 d^2+11 d+12$
We are given the quadratic expression \(2d^2 + 11d + 12\), which is in the standard form \(Ad^2 + Bd + C\) where \(A=2\), \(B=11\), and \(C=12\). Our objective is to factor this expression into the product of two linear factors. The AC method (or grouping method) is a reliable approach for this type of problem. It involves finding two numbers that multiply to \(A \times C\) and add up to \(B\).
First, we calculate the product of the leading coefficient \(A=2\) and the constant term \(C=12\), which gives \(A \times C = 2 \times 12 = 24\). Next, we search for two integers, \(m\) and \(n\), such that their product \(m \times n = 24\) and their sum \(m + n = 11\). Testing pairs of factors of \(24\), we find that \(3 \times 8 = 24\) and \(3 + 8 = 11\). Thus, our two numbers are 3 and 8.
Now, we rewrite the middle term \(11d\) in the original expression as \(3d + 8d\). The expression becomes:
\[2d^2 + 3d + 8d + 12\]
We can now group the terms to facilitate factoring:
\[(2d^2 + 3d) + (8d + 12)\]
Next, we factor out the greatest common factor (GCF) from each group:
The expression is now \(d(2d + 3) + 4(2d + 3)\).
Finally, since both terms share a common binomial factor of \((2d+3)\), we can factor it out:
\[(d + 4)(2d + 3)\]
Thus, the expression \(2d^2 + 11d + 12\) is factored into \((d + 4)(2d + 3)\). To verify, we can expand it: \(d(2d) + d(3) + 4(2d) + 4(3) = 2d^2 + 3d + 8d + 12 = 2d^2 + 11d + 12\), which confirms our result. The completely factored expression is \(\boxed{(d+4)(2d+3)}\).
Factoring quadratic expressions is a fundamental skill in algebra used for solving equations, graphing parabolas, and simplifying rational expressions. For instance, if you are designing a rectangular garden with an area represented by the quadratic \(2d^2 + 11d + 12\), finding the factors allows you to determine the expressions for the length and width of the garden as \((d+4)\) and \((2d+3)\) respectively, helping you in planning the dimensions relative to a variable \(d\).