Question

If $\alpha$ and $\beta$ are are zeroes of $p\left(x\right)=x^2-2x-19$, find a polynomial zeroes $2\alpha$ and $2\beta$.

• A. $x^2-4x+76=0$
• B. $x^2-4x-76=0$
• C. $x^2+4x+76=0$
• D. $x^2+4x-76=0$

Answer 1

Answer: (B)

$x^2-4x-76=0$

If $m$ and $n$ are the roots of the quadratic equation $A{x}^2+Bx+C=0$, then

$m+n=-\frac{B}{A}$

$mn=\frac{C}{A}$

and the equation can be written in the form $x^2-(m+n)x+mn=0$

$\alpha$ and $\beta$ are the roots of the quadratic equation,

$x^2-2x-19=0$

So, for the given equation,

$\alpha +\beta =2$........(1)

$\alpha \beta =-19$............(2)

If $2\alpha$ and $2\beta$ are the roots of a quadratic equation, then the equation is

$x^2-(2\alpha +2\beta )x+\left(2\alpha \right)\left(2\beta \right)=0$

$x^2-2(\alpha +\beta )x+4\left(\alpha \right)\left(\beta \right)=0$

$x^2-2\left(2\right)x+4(-19)=0$

$x^2-4x-76=0$