Question

If $\alpha ,\beta$ are the roots of the quadratic equation $x^2-7x+12=0$. Then the equation with roots $-\alpha ,-\beta$ is

• A. $x^2-7x-12=0$
• B. $x^2+7x+12=0$
• C. $x^2+7x-12=0$
• D. $x^2-7x+12=0$

Answer 1

The correct option is B $x^2+7x+12=0$

Given: $x^2-7x+12=0$ with roots $\alpha ,\beta$
To find: Quadratic equation with roots $-\alpha ,-\beta$

Sum of roots $\alpha +\beta =-\frac{b}{a}=-\frac{-7}{1}=7$

Product of roots $\alpha \cdot \beta =\frac{c}{a}=\frac{12}{1}=12$

Now, quadratic equation with roots $-\alpha ,-\beta$ will have:
Sum of roots: $-(\alpha +\beta )=-7$
Product of roots: $(-\alpha )\cdot (-\beta )=12$

Hence the transformed quadratic equation with roots $-\alpha ,-\beta$ will be

$\Rightarrow x^2+(\alpha +\beta )x+\alpha .\beta =0$

$\Rightarrow x^2+7x+12=0$