Question

What are the solutions of the polynomial function f(x)=x^2−12x+35

Answer 1

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:

1.
\[
x = \frac{12 + 2}{2} = \frac{14}{2} = 7
\]

2.
\[
x = \frac{12 - 2}{2} = \frac{10}{2} = 5
\]

Thus, the solutions of the polynomial function \( f(x) = x^2 - 12x + 35 \) are \( x = 7 \) and \( x = 5 \).