Alright, let's break down the given problem and solution step by step.
You've started with the quadratic equation:
[tex]\[ 3x^2 + 17x - 6 = 0 \][/tex]
You then factored the quadratic equation:
[tex]\[ 3x^2 + 18x - x - 6 = 0 \][/tex]
Grouping the terms to factor by grouping:
[tex]\[ 3x(x + 6) - 1(x + 6) = 0 \][/tex]
Factoring out the common term [tex]\((x + 6)\)[/tex]:
[tex]\[ (3x - 1)(x + 6) = 0 \][/tex]
Next, we solve each binomial set to zero:
[tex]\[ 3x - 1 = 0 \][/tex]
[tex]\[ 3x = 1 \][/tex]
[tex]\[ x = \frac{1}{3} \][/tex]
[tex]\[ x + 6 = 0 \][/tex]
[tex]\[ x = -6 \][/tex]
So, the solutions to the equation [tex]\(3x^2 + 17x - 6 = 0\)[/tex] are:
[tex]\[ x = \frac{1}{3} \quad \text{and} \quad x = -6 \][/tex]
Now, let’s see which values from the list are the solutions to our quadratic equation:
- [tex]\(x = -6\)[/tex] (Yes, it is a solution as derived)
- [tex]\(x \rightarrow 6\)[/tex] (No, this was not a solution we found)
- [tex]\(x = -\frac{1}{3}\)[/tex] (No, this was not a solution we found)
- [tex]\(x = \frac{1}{3}\)[/tex] (Yes, it is a solution as derived)
- [tex]\(x = 0\)[/tex] (No, this was not a solution we found)
Therefore, the possible solutions of the equation [tex]\(3x^2 + 17x - 6 = 0\)[/tex] are:
[tex]\[ x = -6 \][/tex]
[tex]\[ x = \frac{1}{3} \][/tex]
So, the correct boxes to check are:
- [tex]\(x = -6\)[/tex]
- [tex]\(x = \frac{1}{3}\)[/tex]
