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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2022 · Mathematics · Quadratic Equation

If the roots of the equation x^2-a x-b x-c x+b c+c a=0 are equal,...

If the roots of the equation x2−ax−bx−cx+bc+ca=0x^2-a x-b x-c x+b c+c a=0 are equal, then which one of the following is correct?

Options

  1. Aa+b+c=0a+b+c=0
  2. Ba−b+c=0a-b+c=0
  3. Ca+b−c=0a+b-c=0
  4. D−a+b+c=0-a+b+c=0

Correct answer

C. a+b−c=0a+b-c=0

Step-by-step solution

Since x2−(a+b+c)x+bc+ca=0x^2-(a+b+c) x+b c+c a=0 has equal roots ∴[−(a+b+c)]2−4.1(bc+ca)=0⇒[(a+b)+c]2−4c(a+b)=0⇒(a+b)2+c2+2c(a+b)−4c(a+b)=0⇒(a+b−c)2=0⇒a+b−c=0\begin{aligned} & \therefore \quad[-(a+b+c)]^2-4.1(b c+c a)=0 \\ & \Rightarrow \quad[(a+b)+c]^2-4 c(a+b)=0 \\ & \Rightarrow \quad(a+b)^2+c^2+2 c(a+b)-4 c(a+b)=0 \\ & \Rightarrow \quad(a+b-c)^2=0 \\ & \Rightarrow a+b-c=0\end{aligned}
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