Let's solve the quadratic equation [tex]\( g(x) = 6x^2 + 23x - 4 = 0 \)[/tex].
To solve the quadratic equation, we will use the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 6 \)[/tex]
- [tex]\( b = 23 \)[/tex]
- [tex]\( c = -4 \)[/tex]
First, we calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = 23^2 - 4 \cdot 6 \cdot (-4) \][/tex]
[tex]\[ \Delta = 529 + 96 \][/tex]
[tex]\[ \Delta = 625 \][/tex]
Next, we find the square root of the discriminant:
[tex]\[ \sqrt{625} = 25 \][/tex]
Now, we use the quadratic formula to find the roots:
[tex]\[ x = \frac{-23 \pm 25}{2 \cdot 6} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{-23 + 25}{12} = \frac{2}{12} = \frac{1}{6} \][/tex]
and
[tex]\[ x_2 = \frac{-23 - 25}{12} = \frac{-48}{12} = -4 \][/tex]
So, the solutions to the equation [tex]\( g(x) = 0 \)[/tex] are:
[tex]\[ x = \frac{1}{6} \quad \text{or} \quad x = -4 \][/tex]
Now, let's compare these solutions with the provided options:
1. [tex]\( x = -6 \)[/tex] or [tex]\( x = \frac{1}{4} \)[/tex]
2. [tex]\( x = -4 \)[/tex] or [tex]\( x = \frac{1}{6} \)[/tex]
3. [tex]\( x = -\frac{1}{4} \)[/tex] or [tex]\( x = 6 \)[/tex]
4. [tex]\( x = -\frac{1}{6} \)[/tex] or [tex]\( x = 4 \)[/tex]
By matching the solutions, we see that the correct option is:
[tex]\[ x = -4 \quad \text{or} \quad x = \frac{1}{6} \][/tex]
Therefore, the correct answer is:
Option 2: [tex]\( x = -4 \)[/tex] or [tex]\( x = \frac{1}{6} \)[/tex]
