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**Seaniboy** The roots of the quadratic equation$\displaystyle x_2 + (x + 1) = k$ are $\displaystyle {\alpha}+{\beta}$

(i) prove that $\displaystyle {\alpha}_2 + {\beta}_2 = k$;

$\displaystyle 2x_2 + 2x + 1 - k = 0$

$\displaystyle {\alpha} + {\beta} =\frac{-b}{a}= -1$; $\displaystyle {\alpha}{\beta} = \frac{c}{a} = \frac{1-k}{2}$

$\displaystyle {\alpha}_2 + {\beta}_2 = ({\alpha} + {\beta})_2 - 2{\alpha}{\beta} = (-1)_2 + 2(\frac{1 - k}{2}) = -1 + 1 + k = k$; Hence $\displaystyle {\alpha}_2 + {\beta}_2 = k$

(ii) Find, in terms of $\displaystyle k$, a quadratic equation in $\displaystyle x$, whose roots are $\displaystyle {\alpha}_2$ and $\displaystyle {\beta}_2$.

Formula for quadratic equation where $\displaystyle {\alpha}$ and $\displaystyle {\beta}$ are the roots of the equation $\displaystyle ax_2 + bx + c = 0$:

$\displaystyle x_2$ - sum of roots + product of roots;

Where the equation's roots are $\displaystyle {\alpha}_2$ and $\displaystyle {\beta}_2$: we know from (i) above that $\displaystyle {\alpha}_2 + {\beta}_2 = k$ and that $\displaystyle {\alpha}{\beta} = \frac{1 - k}{2}$;

so $\displaystyle x_2$ - sum of roots + product of roots = 0 implies $\displaystyle 2x_2 - 2kx + (\frac{1 - k}{2})_2$

According to the textbook the answer is $\displaystyle 4x_2 - 4kx + (1 - k)_2 = 0$