Question:
If \(\alpha\) and \(\beta\) are the roots of the equation \(ax^2 + bx + c = 0\), where \(a \neq 0\), then \((a\alpha + b)(a\beta + b)\) is equal to
यदि \(\alpha\) और \(\beta\) समीकरण \(ax^2 + bx + c = 0\) के मूल हैं, जहाँ \(a \neq 0\) है, तो \((a\alpha + b)(a\beta + b)\) किसके बराबर है?
Options:
- ab
- abc
- ca
- bc
- ab
- abc
- ca
- bc
Correct Answer: c)
Explanation:
\((a\alpha + b)(a\beta + b) = a^2\alpha\beta + ab(\alpha+\beta) + b^2\).
Substituting \(\alpha\beta = \dfrac{c}{a}\) and \(\alpha+\beta = -\dfrac{b}{a}\):
\(= a^2\left(\dfrac{c}{a}\right) + ab\left(-\dfrac{b}{a}\right) + b^2 = ac – b^2 + b^2 = ac = ca\).
\((a\alpha + b)(a\beta + b) = a^2\alpha\beta + ab(\alpha+\beta) + b^2\)।
\(\alpha\beta = \dfrac{c}{a}\) तथा \(\alpha+\beta = -\dfrac{b}{a}\) प्रतिस्थापित करने पर:
\(= a^2(c/a) + ab(-b/a) + b^2 = ac – b^2 + b^2 = ac = ca\)।