Question

If \(α, β\) are the roots of a quadratic equation \(ax^2+bx+c=0,\ a≠0\) then \(α^2+β^2\)

• A. \(\frac {1}{b^2}(a^2+2bc)\)
• B. \(\frac {1}{b^2}(a^2-2bc)\)
• C. \(\frac {1}{a^2}(b^2+2ac)\)
• D. \(\frac {1}{a^2}(b^2-2ac)\)

Answer 1

The Correct Option is D

To solve the problem, we need to find the value of \( \alpha^2 + \beta^2 \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

1. Understanding the Identity:
We use the identity:

\( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \)

2. Sum and Product of Roots:
For the quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots are:

\( \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \)

3. Apply the Identity:
Substitute the values into the identity:

\[ \alpha^2 + \beta^2 = \left( -\frac{b}{a} \right)^2 - 2 \cdot \frac{c}{a} = \frac{b^2}{a^2} - \frac{2c}{a} \]

4. Expressing with Common Denominator:
We take the LCM of the terms to combine them:

\[ \alpha^2 + \beta^2 = \frac{b^2 - 2ac}{a^2} \]

Final Answer:
The value of \( \alpha^2 + \beta^2 \) is \( \frac{b^2 - 2ac}{a^2} \).