To solve for [tex]\(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\)[/tex] where [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are the roots of the polynomial [tex]\(f(x) = ax^2 + bx + c\)[/tex], we can use the properties of the roots of polynomials and some algebraic manipulations.
Let's start by recalling the fundamental relationships between the roots and coefficients of a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex]:
1. The sum of the roots, [tex]\(\alpha + \beta\)[/tex], is given by [tex]\(-\frac{b}{a}\)[/tex].
2. The product of the roots, [tex]\(\alpha \beta\)[/tex], is given by [tex]\(\frac{c}{a}\)[/tex].
We need to find [tex]\(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\)[/tex].
Firstly, let's express [tex]\(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\)[/tex] in a different form:
[tex]\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\beta^2 + \alpha^2}{\alpha^2 \beta^2} \][/tex]
Now, let's use the relationship between the sum and product of the roots:
[tex]\[ \alpha^2 \beta^2 = (\alpha \beta)^2 = \left( \frac{c}{a} \right)^2 = \frac{c^2}{a^2} \][/tex]
Next, we need to express [tex]\(\alpha^2 + \beta^2\)[/tex] in terms of [tex]\(\alpha + \beta\)[/tex] and [tex]\(\alpha \beta\)[/tex]. We use the identity:
[tex]\[ (\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2 \][/tex]
From which it follows:
[tex]\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \][/tex]
Substituting the expressions for [tex]\(\alpha + \beta\)[/tex] and [tex]\(\alpha \beta\)[/tex]:
[tex]\[ \alpha^2 + \beta^2 = \left( -\frac{b}{a} \right)^2 - 2 \left( \frac{c}{a} \right) \][/tex]
[tex]\[ \alpha^2 + \beta^2 = \frac{b^2}{a^2} - \frac{2c}{a} \][/tex]
Next, substitute [tex]\(\alpha^2 + \beta^2\)[/tex] and [tex]\(\alpha^2 \beta^2\)[/tex] back into the expression for [tex]\(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\)[/tex]:
[tex]\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\frac{c^2}{a^2}} \][/tex]
Simplify the fraction:
[tex]\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{b^2 - 2ac}{c^2} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{b^2 - 2ac}{c^2}} \][/tex]
Therefore, the answer is option (b) [tex]\(\frac{b^2-2ac}{c^2}\)[/tex].
