To determine which number, if any, is a solution to the equation [tex]\( 5x^3 + 6x^2 + 15x + 18 = 0 \)[/tex], we will substitute each given option into the equation and check if the equation holds true.
Given options:
1. [tex]\( x = \frac{5}{6} \)[/tex]
2. [tex]\( x = -\frac{6}{5} \)[/tex]
3. [tex]\( x = \sqrt{3} \)[/tex]
4. [tex]\( x = -\sqrt{6} \)[/tex]
### Checking [tex]\( x = \frac{5}{6} \)[/tex]:
Substitute [tex]\( x = \frac{5}{6} \)[/tex] into the equation:
[tex]\[
5 \left( \frac{5}{6} \right)^3 + 6 \left( \frac{5}{6} \right)^2 + 15 \left( \frac{5}{6} \right) + 18
\][/tex]
Calculate each term separately:
[tex]\[
5 \left( \frac{125}{216} \right) + 6 \left( \frac{25}{36} \right) + 15 \left( \frac{5}{6} \right) + 18
\][/tex]
[tex]\[
= \frac{625}{216} + \frac{150}{36} + \frac{75}{6} + 18
\][/tex]
[tex]\[
= \frac{625}{216} + \frac{900}{216} + \frac{2700}{216} + \frac{3888}{216}
\][/tex]
[tex]\[
= \frac{625 + 900 + 2700 + 3888}{216} = \frac{8113}{216}
\][/tex]
Clearly, this does not equal zero, so [tex]\( x = \frac{5}{6} \)[/tex] is not a solution.
### Checking [tex]\( x = -\frac{6}{5} \)[/tex]:
Substitute [tex]\( x = -\frac{6}{5} \)[/tex] into the equation:
[tex]\[
5 \left( -\frac{6}{5} \right)^3 + 6 \left( -\frac{6}{5} \right)^2 + 15 \left( -\frac{6}{5} \right) + 18
\][/tex]
Calculate each term separately:
[tex]\[
5 \left( -\frac{216}{125} \right) + 6 \left( \frac{36}{25} \right) + 15 \left( -\frac{6}{5} \right) + 18
\][/tex]
[tex]\[
= -\frac{1080}{125} + \frac{216}{25} - \frac{90}{5} + 18
\][/tex]
[tex]\[
= -\frac{1080}{125} + \frac{540}{125} - \frac{225}{25} + 18
\][/tex]
[tex]\[
= -\frac{1080}{125} + \frac{540}{125} - \frac{900}{125} + \frac{2250}{125}
\][/tex]
Simplifying:
[tex]\[
= \frac{-1080 + 540 - 900 + 2250}{125} = \frac{810}{125} = 6.48
\][/tex]
Clearly, this does not equal zero, so [tex]\( x = -\frac{6}{5} \)[/tex] is not a solution.
### Checking [tex]\( x = \sqrt{3} \)[/tex]:
Substitute [tex]\( x = \sqrt{3} \)[/tex] into the equation:
[tex]\[
5 (\sqrt{3})^3 + 6 (\sqrt{3})^2 + 15 (\sqrt{3}) + 18
\][/tex]
Calculate each term separately:
[tex]\[
5 (3\sqrt{3}) + 6 (3) + 15 (\sqrt{3}) + 18
\][/tex]
[tex]\[
= 15\sqrt{3} + 18 + 15\sqrt{3} + 18
\][/tex]
[tex]\[
= 30\sqrt{3} + 36
\][/tex]
Clearly, this does not equal zero, so [tex]\( x = \sqrt{3} \)[/tex] is not a solution.
### Checking [tex]\( x = -\sqrt{6} \)[/tex]:
Substitute [tex]\( x = -\sqrt{6} \)[/tex] into the equation:
[tex]\[
5 (-\sqrt{6})^3 + 6 (-\sqrt{6})^2 + 15 (-\sqrt{6}) + 18
\][/tex]
Calculate each term separately:
[tex]\[
5 (-6\sqrt{6}) + 6 (6) + 15 (-\sqrt{6}) + 18
\][/tex]
[tex]\[
= -30\sqrt{6} + 36 - 15\sqrt{6} + 18
\][/tex]
[tex]\[
= -45\sqrt{6} + 54
\][/tex]
Clearly, this does not equal zero, so [tex]\( x = -\sqrt{6} \)[/tex] is not a solution.
Upon testing all given options, we find that none of them satisfy the equation [tex]\( 5x^3 + 6x^2 + 15x + 18 = 0 \)[/tex].
Therefore, none of the given numbers are solutions to the equation.
