To find the factored form of the quadratic equation [tex]\(6x^2 + 13x + 6\)[/tex], we need to express it as a product of two binomials. We can write it as follows:
[tex]\[ 6x^2 + 13x + 6 = (a x + b)(c x + d) \][/tex]
We need to determine the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] such that when we expand the product, we get the original quadratic expression.
1. Identify the constants and coefficients:
[tex]\[ ax \cdot cx = 6x^2 \][/tex]
[tex]\[ ad + bc = 13x \][/tex]
[tex]\[ b \cdot d = 6 \][/tex]
2. List possible pairs of factors:
For [tex]\(6x^2\)[/tex], the pairs could be [tex]\((2x \cdot 3x)\)[/tex].
For [tex]\(6\)[/tex], the pairs could be [tex]\((1 \cdot 6)\)[/tex], [tex]\((2 \cdot 3)\)[/tex].
3. Find the suitable pairs of constants that satisfy the middle term:
Let's try [tex]\((2x + 3)\)[/tex] and [tex]\((3x + 2)\)[/tex]:
[tex]\[
(2x + 3)(3x + 2)
\][/tex]
4. Expand to verify:
[tex]\[
(2x + 3)(3x + 2) = 2x \cdot 3x + 2x \cdot 2 + 3 \cdot 3x + 3 \cdot 2
\][/tex]
[tex]\[
= 6x^2 + 4x + 9x + 6
\][/tex]
[tex]\[
= 6x^2 + 13x + 6
\][/tex]
Thus, the binomials [tex]\((2x + 3)\)[/tex] and [tex]\((3x + 2)\)[/tex] are the factors of the quadratic equation [tex]\(6x^2 + 13x + 6\)[/tex].
Therefore, the factored form of [tex]\(6x^2 + 13x + 6\)[/tex] is:
[tex]\[ (2x + 3)(3x + 2) \][/tex]
So, the correct answer is:
[tex]\[ (2x + 3)(3x + 2) \][/tex]
