Question

lf the roots of the equation $a{x}^2+bx+c=0$ are the same , when each of the coefficient is increased by the same quantity, then $x^2+x+1$ is equal to

• A. $1$
• B. $0$
• C. $3$
• D. $2$

Answer 1

The correct option is B $0$

Let the coefficients of the given equation be increased by $k$. The new equation is:
$(a+k)x^2+(b+k)x+(c+k)=0$
Since the roots are unchanged, the coefficient of $x^2$, $x$ and $c$ in the original and new equation are in proportion.
$\therefore \frac{a+k}{a}=\frac{b+k}{b}=\frac{c+k}{c}$
$\Rightarrow \frac{a+k-a}{a}=\frac{b+k-b}{b}=\frac{c+k-c}{c}$
$\Rightarrow \frac{k}{a}=\frac{k}{b}=\frac{k}{c}$
$\Rightarrow a=b=c$
The original equation becomes $a{x}^2+ax+a=0$
$\Rightarrow x^2+x+1=0$
Ans- Option B