To find the x-intercepts of the function \( f(x) = 2x^2 - 11x + 14 \), we need to solve the equation \( f(x) = 0 \).
This means we need to set the equation equal to zero and solve for \( x \):
\[ 2x^2 - 11x + 14 = 0 \]
Now, we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -11 \), and \( c = 14 \).
- Calculate the discriminant:
\[ b^2 - 4ac = (-11)^2 - 4 \cdot 2 \cdot 14 \] \[ = 121 - 112 = 9 \]
- Now substitute into the quadratic formula:
\[ x = \frac{-(-11) \pm \sqrt{9}}{2 \cdot 2} \] \[ x = \frac{11 \pm 3}{4} \]
- This gives us two possible solutions:
- For the positive case:
\[ x = \frac{11 + 3}{4} = \frac{14}{4} = 3.5 \]
- For the negative case:
\[ x = \frac{11 - 3}{4} = \frac{8}{4} = 2 \]
Thus, the x-intercepts of the function \( f(x) = 2x^2 - 11x + 14 \) are:
\[ x = 2 \quad \text{and} \quad x = 3.5 \]
So the x-intercepts are \( x = 2 \) and \( x = 3.5 \).
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