To solve the given equation [tex]\(3x^2 - 18x + 5 = 47\)[/tex], we will begin by simplifying and solving the quadratic equation step-by-step.
1. First, rewrite the equation in standard quadratic form:
[tex]\[
3x^2 - 18x + 5 = 47
\][/tex]
Subtract 47 from both sides to set the equation to 0:
[tex]\[
3x^2 - 18x + 5 - 47 = 0
\][/tex]
Simplify the constants:
[tex]\[
3x^2 - 18x - 42 = 0
\][/tex]
2. Identify the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
a = 3, \quad b = -18, \quad c = -42
\][/tex]
3. Calculate the discriminant ([tex]\(\Delta\)[/tex]) to determine the nature of the roots:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plug in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-18)^2 - 4 \cdot 3 \cdot (-42)
\][/tex]
[tex]\[
\Delta = 324 + 504 = 828
\][/tex]
4. Use the quadratic formula to find the roots:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(b = -18\)[/tex], [tex]\(a = 3\)[/tex] and [tex]\(\Delta = 828\)[/tex]:
[tex]\[
x = \frac{18 \pm \sqrt{828}}{6}
\][/tex]
5. Simplify the expression [tex]\(\sqrt{828}\)[/tex]:
[tex]\[
\sqrt{828} = \sqrt{4 \cdot 207} = 2\sqrt{207}
\][/tex]
Therefore:
[tex]\[
x = \frac{18 \pm 2\sqrt{207}}{6}
\][/tex]
6. Simplify further by factoring out common terms:
[tex]\[
x = \frac{18}{6} \pm \frac{2\sqrt{207}}{6}
\][/tex]
[tex]\[
x = 3 \pm \frac{\sqrt{207}}{3}
\][/tex]
Since [tex]\(\sqrt{207}\)[/tex] simplifies directly and it's not a simplification of [tex]\(\sqrt{51}\)[/tex], we can recognize the form:
[tex]\(
x = 3 \pm \sqrt{51}
\)[/tex]
Thus, the exact roots are:
[tex]\[
x = 3 + \sqrt{51} \quad \text{and} \quad x = 3 - \sqrt{51}
\][/tex]
Given the multiple choice answers, the correct solution is:
[tex]\[
x = 3 \pm \sqrt{51}
\][/tex]
