To solve the polynomial equation [tex]\(0 = x^2 - 18x + 65\)[/tex], we need to find the roots of the equation, which are the values of [tex]\(x\)[/tex] that satisfy the equation.
The polynomial equation given is:
[tex]\[x^2 - 18x + 65 = 0\][/tex]
We can use the quadratic formula to find the roots of this quadratic equation. The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
In this equation, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term, respectively. For the given equation [tex]\(x^2 - 18x + 65\)[/tex]:
[tex]\[a = 1, \quad b = -18, \quad c = 65\][/tex]
Substituting these values into the quadratic formula:
[tex]\[x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 65}}{2 \cdot 1}\][/tex]
[tex]\[x = \frac{18 \pm \sqrt{324 - 260}}{2}\][/tex]
[tex]\[x = \frac{18 \pm \sqrt{64}}{2}\][/tex]
[tex]\[x = \frac{18 \pm 8}{2}\][/tex]
This results in two solutions:
1. [tex]\(x = \frac{18 + 8}{2} = \frac{26}{2} = 13\)[/tex]
2. [tex]\(x = \frac{18 - 8}{2} = \frac{10}{2} = 5\)[/tex]
Therefore, the values that are the solutions to the polynomial equation [tex]\(0 = x^2 - 18x + 65\)[/tex] are:
[tex]\[x = 5 \quad \text{and} \quad x = 13\][/tex]
So, the correct answers are:
5 and 13
