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GATE 2020 EC – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

$a,b,c$ are real numbers. The quadratic equation $ax^2-bx+c=0$ has equal roots, which is $\beta$, then

  1. $\beta=b/a$
  2. $\beta^2=ac$
  3. $\beta^3=bc/(2a^2)$
  4. $b^2\ne4ac$

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Show answer and explanation

Correct answer: (C) $\beta^3=bc/(2a^2)$

Explanation

Equal roots need $b^2=4ac$, and the root is $\beta=\frac{b}{2a}$. Then $\beta^3=\frac{b^3}{8a^3}=\frac{b\cdot4ac}{8a^3}=\frac{bc}{2a^2}$.