Sure, let's solve the problem step-by-step:
1. Identify the roots of the quadratic equation [tex]\(x^2 - 5x + 4\)[/tex].
For the quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], the roots are given by the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In this case, the coefficients are [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 4\)[/tex].
2. Calculate the roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
[tex]\[
\alpha = \frac{-(-5) + \sqrt{(-5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{5 + \sqrt{25 - 16}}{2} = \frac{5 + \sqrt{9}}{2} = \frac{5 + 3}{2} = 4
\][/tex]
[tex]\[
\beta = \frac{-(-5) - \sqrt{(-5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{5 - \sqrt{25 - 16}}{2} = \frac{5 - \sqrt{9}}{2} = \frac{5 - 3}{2} = 1
\][/tex]
So, the roots are [tex]\(\alpha = 4\)[/tex] and [tex]\(\beta = 1\)[/tex].
3. Find the value of [tex]\(\frac{1}{\alpha} + \frac{1}{\beta}\)[/tex].
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{1}{4} + \frac{1}{1} = \frac{1}{4} + 1 = \frac{1}{4} + \frac{4}{4} = \frac{5}{4} = 1.25
\][/tex]
4. Calculate the product [tex]\(\alpha \cdot \beta\)[/tex].
[tex]\[
\alpha \cdot \beta = 4 \cdot 1 = 4
\][/tex]
5. Substitute these values into the given expression:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} - 2 \cdot \alpha \cdot \beta
\][/tex]
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} - 2 \cdot \alpha \beta = 1.25 - 2 \cdot 4 = 1.25 - 8 = -6.75
\][/tex]
Thus, the value of [tex]\(\frac{1}{\alpha} + \frac{1}{\beta} - 2 \cdot \alpha \beta\)[/tex] is [tex]\(-6.75\)[/tex].
