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To find the value of [tex]\(\frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2}\)[/tex] given that [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are the roots of the quadratic equation [tex]\(a x^2 + b x + c = 0\)[/tex], let's walk through the process step-by-step.
1. Identify the roots of the quadratic equation:
[tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are roots of the equation [tex]\(a x^2 + b x + c = 0\)[/tex]. Therefore, they satisfy the equation:
[tex]\[ a \alpha^2 + b \alpha + c = 0 \][/tex]
[tex]\[ a \beta^2 + b \beta + c = 0 \][/tex]
2. Express the quadratic equation in factored form:
The quadratic equation can be written as:
[tex]\[ a (x - \alpha) (x - \beta) = 0 \][/tex]
Expanding this, we get:
[tex]\[ ax^2 - a(\alpha + \beta)x + a \alpha \beta = 0 \][/tex]
3. Comparing coefficients:
By comparing the coefficients with the standard form [tex]\(a x^2 + b x + c = 0\)[/tex], we get:
[tex]\[ -a(\alpha + \beta) = b \implies \alpha + \beta = -\frac{b}{a} \][/tex]
[tex]\[ a \alpha \beta = c \implies \alpha \beta = \frac{c}{a} \][/tex]
4. Form the initial expression:
We need to evaluate:
[tex]\[ \frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2} \][/tex]
5. Utilize the properties of roots:
From the properties of the roots,
[tex]\[ a \alpha^2 + b \alpha + c = 0 \implies a \alpha^2 = -b \alpha - c \implies a \alpha^2 + b \alpha = -c \][/tex]
This helps in simplifying [tex]\(a \alpha + b\)[/tex]:
[tex]\[ a \alpha + b = -\frac{c}{\alpha} \][/tex]
Similarly,
[tex]\[ a \beta + b = -\frac{c}{\beta} \][/tex]
6. Substitute these expressions into the target expression:
Substitute [tex]\(a \alpha + b = -\frac{c}{\alpha}\)[/tex] and [tex]\(a \beta + b = -\frac{c}{\beta}\)[/tex]:
[tex]\[ \frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2} = \frac{1}{\left(-\frac{c}{\alpha}\right)^2} + \frac{1}{\left(-\frac{c}{\beta}\right)^2} \][/tex]
Simplifying,
[tex]\[ \frac{\alpha^2}{c^2} + \frac{\beta^2}{c^2} = \frac{\alpha^2 + \beta^2}{c^2} \][/tex]
7. Simplify the terms [tex]\(\alpha^2 + \beta^2\)[/tex]:
Using the identity [tex]\(\alpha + \beta\)[/tex]^2 = \alpha^2 + \beta^2 + 2\alpha\beta,
[tex]\[ (\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha \beta \implies \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \][/tex]
Substitute [tex]\(\alpha + \beta = -\frac{b}{a}\)[/tex] and [tex]\(\alpha \beta = \frac{c}{a}\)[/tex],
[tex]\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} \][/tex]
8. Final substitution:
Substitute [tex]\(\alpha^2 + \beta^2\)[/tex] back into [tex]\(\frac{\alpha^2 + \beta^2}{c^2}\)[/tex],
[tex]\[ \frac{b^2/a^2 - 2c/a}{c^2} = \frac{b^2 - 2ac}{a^2 c^2} = \frac{b^2 - 2ac}{a^2 c^2} \][/tex]
So the value is:
[tex]\[ \frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2} = (a \beta + b)^{-2} + (a \alpha + b)^{-2} \][/tex]
This completes our detailed process.
1. Identify the roots of the quadratic equation:
[tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are roots of the equation [tex]\(a x^2 + b x + c = 0\)[/tex]. Therefore, they satisfy the equation:
[tex]\[ a \alpha^2 + b \alpha + c = 0 \][/tex]
[tex]\[ a \beta^2 + b \beta + c = 0 \][/tex]
2. Express the quadratic equation in factored form:
The quadratic equation can be written as:
[tex]\[ a (x - \alpha) (x - \beta) = 0 \][/tex]
Expanding this, we get:
[tex]\[ ax^2 - a(\alpha + \beta)x + a \alpha \beta = 0 \][/tex]
3. Comparing coefficients:
By comparing the coefficients with the standard form [tex]\(a x^2 + b x + c = 0\)[/tex], we get:
[tex]\[ -a(\alpha + \beta) = b \implies \alpha + \beta = -\frac{b}{a} \][/tex]
[tex]\[ a \alpha \beta = c \implies \alpha \beta = \frac{c}{a} \][/tex]
4. Form the initial expression:
We need to evaluate:
[tex]\[ \frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2} \][/tex]
5. Utilize the properties of roots:
From the properties of the roots,
[tex]\[ a \alpha^2 + b \alpha + c = 0 \implies a \alpha^2 = -b \alpha - c \implies a \alpha^2 + b \alpha = -c \][/tex]
This helps in simplifying [tex]\(a \alpha + b\)[/tex]:
[tex]\[ a \alpha + b = -\frac{c}{\alpha} \][/tex]
Similarly,
[tex]\[ a \beta + b = -\frac{c}{\beta} \][/tex]
6. Substitute these expressions into the target expression:
Substitute [tex]\(a \alpha + b = -\frac{c}{\alpha}\)[/tex] and [tex]\(a \beta + b = -\frac{c}{\beta}\)[/tex]:
[tex]\[ \frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2} = \frac{1}{\left(-\frac{c}{\alpha}\right)^2} + \frac{1}{\left(-\frac{c}{\beta}\right)^2} \][/tex]
Simplifying,
[tex]\[ \frac{\alpha^2}{c^2} + \frac{\beta^2}{c^2} = \frac{\alpha^2 + \beta^2}{c^2} \][/tex]
7. Simplify the terms [tex]\(\alpha^2 + \beta^2\)[/tex]:
Using the identity [tex]\(\alpha + \beta\)[/tex]^2 = \alpha^2 + \beta^2 + 2\alpha\beta,
[tex]\[ (\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha \beta \implies \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \][/tex]
Substitute [tex]\(\alpha + \beta = -\frac{b}{a}\)[/tex] and [tex]\(\alpha \beta = \frac{c}{a}\)[/tex],
[tex]\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} \][/tex]
8. Final substitution:
Substitute [tex]\(\alpha^2 + \beta^2\)[/tex] back into [tex]\(\frac{\alpha^2 + \beta^2}{c^2}\)[/tex],
[tex]\[ \frac{b^2/a^2 - 2c/a}{c^2} = \frac{b^2 - 2ac}{a^2 c^2} = \frac{b^2 - 2ac}{a^2 c^2} \][/tex]
So the value is:
[tex]\[ \frac{1}{(a \alpha + b)^2} + \frac{1}{(a \beta + b)^2} = (a \beta + b)^{-2} + (a \alpha + b)^{-2} \][/tex]
This completes our detailed process.
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