Question

Which of the following statements regarding quadratic equations are true? (A) Solution set for the equation $6x^2 - 5x = 4$ is $\{-\frac{1}{2}, \frac{4}{3}\}$ (B) The nature of the roots of the equation $9x^2 + 6x + 1 = 0$ is equal, real and rational. (C) If one root of $4x^2 - 3x + K = 0$ is 3 times the other, then $K = \frac{27}{256}$

Accepted Answer

B. (A) $6x^2 - 5x - 4 = 0 \implies (2x+1)(3x-4)=0 \implies x = -1/2, 4/3$. True. (B) $9x^2 + 6x + 1 = (3x+1)^2 = 0$. Discriminant $D = 36 - 36 = 0$. Roots are equal, real, and rational ($-1/3, -1/3$). True. (C) Let roots be $r$ and $3r$. Sum of roots = $4r = 3/4 \implies r = 3/16$. Product of roots = $3r^2 = K/4 \implies K = 12 r^2 = 12 \left(\frac{3}{16}ight)^2 = 12 	imes \frac{9}{256} = \frac{27}{64} 
eq \frac{27}{256}$. False. Hence, only (A) and (B) are true.

Truth of quadratic equations statements

Explanation

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