To factor the trinomial [tex]\(6x^2 - 17x + 3\)[/tex], we'll proceed with the following steps:
### Step 1: Identify Coefficients
First, identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the trinomial [tex]\(ax^2 + bx + c\)[/tex]:
- [tex]\(a = 6\)[/tex]
- [tex]\(b = -17\)[/tex]
- [tex]\(c = 3\)[/tex]
### Step 2: Calculate the Discriminant
The discriminant for a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-17)^2 - 4 \cdot 6 \cdot 3 = 289 - 72 = 217
\][/tex]
### Step 3: Solving for the Roots
Using the quadratic formula to find the roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] of the equation [tex]\(6x^2 - 17x + 3 = 0\)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[
x = \frac{-(-17) \pm \sqrt{217}}{2 \cdot 6}
\][/tex]
[tex]\[
x = \frac{17 \pm \sqrt{217}}{12}
\][/tex]
This gives us two solutions (roots):
[tex]\[
\alpha = \frac{17 + \sqrt{217}}{12} \approx 2.6442433218880197
\][/tex]
[tex]\[
\beta = \frac{17 - \sqrt{217}}{12} \approx 0.1890900114453138
\][/tex]
### Step 4: Expressing the Trinomial in Factored Form
From the roots obtained, we can express the trinomial in its factored form. If the roots of the quadratic equation are [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex], the trinomial can be factored as:
[tex]\[
a(x - \alpha)(x - \beta)
\][/tex]
Substituting [tex]\(a = 6\)[/tex], [tex]\(\alpha \approx 2.6442433218880197\)[/tex], and [tex]\(\beta \approx 0.1890900114453138\)[/tex]:
[tex]\[
6(x - 2.6442433218880197)(x - 0.1890900114453138)
\][/tex]
### Conclusion
The trinomial [tex]\(6x^2 - 17x + 3\)[/tex] can be factored as:
[tex]\[
6(x - 2.6442433218880197)(x - 0.1890900114453138)
\][/tex]
