To find the x-intercepts of the function \( f(x) = 3x^2 - 11x - 4 \), we need to determine where \( f(x) = 0 \). This means we need to solve the equation:
\[ 3x^2 - 11x - 4 = 0 \]
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -11 \), and \( c = -4 \).
- Calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 - 4ac = (-11)^2 - 4 \cdot 3 \cdot (-4) = 121 + 48 = 169 \]
- Now, substitute into the quadratic formula:
\[ x = \frac{-(-11) \pm \sqrt{169}}{2 \cdot 3} \]
\[ x = \frac{11 \pm 13}{6} \]
This gives us two solutions:
- \( x = \frac{11 + 13}{6} = \frac{24}{6} = 4 \)
- \( x = \frac{11 - 13}{6} = \frac{-2}{6} = -\frac{1}{3} \)
Thus, the x-intercepts are:
- The negative x-intercept is at \( \left(-\frac{1}{3}, 0\right) \).
- The positive x-intercept is at \( (4, 0) \).
So, the final answers are:
- The negative x-intercept is at \( \left(-\frac{1}{3}, 0\right) \).
- The positive x-intercept is at \( (4, 0) \).