To determine which of the given options are solutions to the equation [tex]\( 3x^2 + 27x + 54 = 0 \)[/tex], we need to solve the quadratic equation.
Step 1: Write down the quadratic equation:
[tex]\[ 3x^2 + 27x + 54 = 0 \][/tex]
Step 2: We can solve this using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 3 \)[/tex], [tex]\( b = 27 \)[/tex], and [tex]\( c = 54 \)[/tex].
Step 3: Calculate the discriminant, [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = 27^2 - 4 \cdot 3 \cdot 54 = 729 - 648 = 81 \][/tex]
Step 4: Find the solutions using the quadratic formula:
[tex]\[ x = \frac{-27 \pm \sqrt{81}}{2 \cdot 3} = \frac{-27 \pm 9}{6} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{-27 + 9}{6} = \frac{-18}{6} = -3 \][/tex]
[tex]\[ x = \frac{-27 - 9}{6} = \frac{-36}{6} = -6 \][/tex]
So, the solutions to the equation are [tex]\( x = -3 \)[/tex] and [tex]\( x = -6 \)[/tex].
Step 5: Match these solutions with the given options:
- Option A: [tex]\( 3 \)[/tex] is not a solution.
- Option B: [tex]\( -3 \)[/tex] is a solution.
- Option C: [tex]\( -6 \)[/tex] is a solution.
- Option D: [tex]\( 9 \)[/tex] is not a solution.
- Option E: [tex]\( 6 \)[/tex] is not a solution.
Therefore, the correct options are:
B. -3
C. -6
