Question

If $a,b,c$ are the roots of the equation $x^3+2x^2+1=0$ . Find the equation whose roots are $b+c-a,c+a-b,a+b-c.$

• A. $x^3+2x^2+4x+9=0$
• B. $x^3$ + $x^2+x+1=0$
• C. $x^3$ + 2 $x^2+x+4=0$
• D. $x^3$ + $x^2+4x+9=0$

Answer 1

The correct option is A

$x^3+2x^2+4x+9=0$

Given: $a,b,c$ are the roots of the equation $x^3+2x^2+1=0$ ___(1)

Using relation between roots and coefficients, we have
$a+b+c=-2$

$(a+b+c)-2a=-2-2a$

$b+c-a=-2-2a$

Let one root is y

$\Rightarrow y=-2-2a$

$\Rightarrow -\frac{(y+2)}{2}=a$

Since $a$ is the roots of the equation
$x^3+2x^2+1=0$

We can replace $a$ by $x$ to generalize it.

- $\frac{(y+2)}{2}=x$

Replace $x$ in terms of $y$ in equation (1)

$\Rightarrow -\frac{(y+2{)}^3}{8}+2\frac{(y+2{)}^2}{4}+1=0$

$\Rightarrow -(y^3+8+6y^2+12y)+4(y^2+2y+4)+1=0$

$\Rightarrow -y^3-2y^2-4y-9=0$
$\Rightarrow y^3+2y^2+4y+9=0$

Write the equation in terms of $x$

$\Rightarrow x^3+2x^2+4x+9=0.$