Question

The equation formed by decreasing the roots of the quadractic equation $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0$, then which of the following statement(s) is/are true?

• A. $a:b:c=1:2:-2$
• B. $a=-c$
• C. $b=-c$
• D. $a:b:c=2:1:-2$

Answer 1

Answer: (A), (C)

$b=-c$

Let the roots of the equation $f\left(x\right)=a{x}^2+bx+c=0$ be $\alpha ,\beta$
Now, the equation with roots $\alpha -1,\beta -1$ is,
$f(x+1)=0\Rightarrow a(x+1{)}^2+b(x+1)+c=0$
$\Rightarrow a{x}^2+(2a+b)x+(a+b+c)=0$
Comparing to the equation
$2x^2+8x+2=0$
So,
$\frac{a}{2}=\frac{(2a+b)}{8}=\frac{a+b+c}{2}=k$
Where is some constant,
$\Rightarrow a=2k\Rightarrow 2a+b=8k\Rightarrow b=4k\Rightarrow a+b+c=2k\Rightarrow c=-4k$

Therefore,
$b=-c,b=2a,c=-2a$
$a:b:c=1:2:-2$