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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2015 · Mathematics · Quadratic Equation

If the sum of the roots of the equation a x^2+b x+c=0 is equal to...

If the sum of the roots of the equation ax2+bx+c=0a x^2+b x+c=0 is equal to the sum of their squares, then

Options

  1. Aa2+b2=c2\mathrm{a}^2+\mathrm{b}^2=\mathrm{c}^2
  2. Ba2+b2=a+ba^2+b^2=a+b
  3. Cab+b2=2ac\mathrm{ab}+\mathrm{b}^2=2 \mathrm{ac}
  4. Dab−b2=2ac\mathrm{ab}-\mathrm{b}^2=2 \mathrm{ac}

Correct answer

C. ab+b2=2ac\mathrm{ab}+\mathrm{b}^2=2 \mathrm{ac}

Step-by-step solution

ax2+bx+c=0a x^2+b x+c=0 Let the root be αandβ\alpha and \beta. α+β=−ba,α⋅β=ca⇒α+β=α2+β2 (given) ⇒α+β=(α+β)2−2αβ⇒−ba=(−ba)2−2ca⇒b2+ab=2ac\begin{aligned} & \alpha+\beta=-\frac{b}{a}, \alpha \cdot \beta=\frac{c}{a} \\ & \Rightarrow \alpha+\beta=\alpha^2+\beta^2 \\ & \text{ (given) } \\ & \Rightarrow \alpha+\beta=(\alpha+\beta)^2-2 \alpha \beta \\ & \Rightarrow-\frac{b}{a}=\left(-\frac{b}{a}\right)^2-\frac{2 c}{a} \\ & \Rightarrow b^2+a b=2 \mathrm{ac} \end{aligned}
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