To find the solutions of the polynomial function \( f(x) = x^2 - 12x + 35 \), we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, the coefficients are:
- \( a = 1 \)
- \( b = -12 \)
- \( c = 35 \)
First, we need to calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-12)^2 - 4 \cdot 1 \cdot 35 \] \[ = 144 - 140 \] \[ = 4 \]
Since the discriminant is positive, there are two distinct real solutions. Now we can substitute the values into the quadratic formula:
\[ x = \frac{-(-12) \pm \sqrt{4}}{2 \cdot 1} \] \[ = \frac{12 \pm 2}{2} \]
This gives us two solutions:
\[ x = \frac{12 + 2}{2} = \frac{14}{2} = 7 \]
\[ x = \frac{12 - 2}{2} = \frac{10}{2} = 5 \]
Thus, the solutions of the polynomial function \( f(x) = x^2 - 12x + 35 \) are \( x = 7 \) and \( x = 5 \).