Question

If a+b+c=0 then roots of eq^n 4ax²+3bx+c=0 are

• A. real and distinct
• B. real and equal
• C. imaginary
• D. can't say

Answer 1

Answer: (D)

can't say

Answer 2

The discriminant of the quadratic equation $4ax^2+3bx+c=0$ is $\Delta = (3b)^2 - 4(4a)(c) = 9b^2 - 16ac$. Given $a+b+c=0$, we have $b = -(a+c)$. Substituting this into the discriminant: $\Delta = 9(-(a+c))^2 - 16ac = 9(a^2 + 2ac + c^2) - 16ac = 9a^2 + 18ac + 9c^2 - 16ac = 9a^2 + 2ac + 9c^2$. We can rewrite $\Delta$ as $9(a + \frac{1}{9}c)^2 + \frac{80}{9}c^2$. If $a \ne 0$ or $c \ne 0$, then $\Delta > 0$, indicating real and distinct roots. If $a=0$, then $b+c=0 \implies c=-b$. The equation becomes $3bx-b=0$. If $b \ne 0$, then $x=1/3$ (a single real root). If $b=0$, then $a=b=c=0$, and the equation is $0=0$, true for all real $x$. Since the nature of the roots can vary, we cannot definitively say they are always real and distinct, real and equal, or imaginary.