Question

If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $4x^2+4x+1$, then find quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.

Answer 1

to solve this lets divide the quadratic equation by 4 such that we have it in the form $x^2+(\alpha +\beta )x+\alpha \beta$

so after dividing by 4 we have

$\frac{4x^2+4x+1}{4}$

$x^2+x+\frac{1}{4}$

So we have$\alpha +\beta =1$ ...................(1)

and$\alpha \beta =\frac{1}{4}$ ..................(2)

So if zeroes to the new quadratic equation are $2\alpha$ and $2\beta$, then

$2\alpha +2\beta =2(\alpha +\beta )$

$=2\left(1\right)=2$ .................... Using (1)

and,

$2\alpha \times 2\beta =4\alpha \beta$

$=4\times \frac{1}{4}=1$ .................... Using (2)

So the new quadratic equation with its roots $2\alpha and2\beta$ will be

$x^2+(2\alpha +2\beta )x+2\alpha \times 2\beta$

Putting the values, the equation would be,

$x^2+2x+1$