For the equation x2 + (2k+2)x + 2k+5 = 0, find the value(s) of k for which the roots are equal. If k>0, find the roots of the new equation whose roots are 2 extalpha + extalpha extbeta and 2 extbeta +

The coefficients are $a=1$, $b=(2k+2)$, and $c=(2k+5)$.

For the roots $\alpha$ and $\beta$ to be equal ($\alpha = \beta$), the discriminant ($\Delta$) of the quadratic equation must be zero.

Divide the entire equation by 4:

Expand the square:

2. (i) (b)

From part (a), the positive value for $k$ is $k=2$.
Substitute $k=2$ into the original quadratic equation to find its roots:

This equation can be factored as $(x+3)^2 = 0$.
So, the roots are $\alpha = -3$ and $\beta = -3$.

The new roots are given as $r_1 = 2\alpha + \alpha\beta$ and $r_2 = 2\beta + \alpha\beta$.
Using $\alpha = -3$ and $\beta = -3$:
First, calculate $\alpha\beta = (-3)(-3) = 9$.

The new roots are 3 and 3.

A quadratic equation with roots $r_1$ and $r_2$ is given by $x^2 - (r