If-and-are-the-roots-of-equation-x-2-px-q-0-and-2-2-are-roots-of-the-equation-x-2-rx-s-0-show-that-the-equation-x-2-4qx-2q-2-r-0-has-real-roots-

Question Number 19945 by Tinkutara last updated on 18/Aug/17

If α and β are the roots of equation

x^{2} + p x + q = 0 and α^{2}, β^{2} are roots of

the equation x^{2} - r x + s = 0, show

that the equation x^{2} - 4 q x + 2 q^{2} - r = 0

has real roots .

Answered by Rasheed.Sindhi last updated on 19/Aug/17

x^{2} + px + q = 0, roots : α, β (given)

x^{2} - rx + s = 0, roots : α^{2}, β^{2} (given)

x^{2} - 4 qx + {2 q}^{2} - r = 0, roots are real . (to be proved)

- - - - - - - - - - -

α + β = - p, α β = q

α^{2} + β^{2} = r, α^{2} β^{2} = s = q^{2}

x^{2} - 4 qx + {2 q}^{2} - r = 0

\Rightarrow x^{2} - 4 (α β) x + 2 {(α β)}^{2} - (α^{2} + β^{2}) = 0

x = \frac{4 α β \pm \sqrt{16 α^{2} β^{2} - 4 (1) (2 {(α β)}^{2} - (α^{2} + β^{2}))}}{2}

x = \frac{4 α β \pm 2 \sqrt{4 α^{2} β^{2} - 2 {(α β)}^{2} + (α^{2} + β^{2})}}{2}

= 2 α β \pm \sqrt{4 α^{2} β^{2} - 2 α^{2} β^{2} + α^{2} + β^{2}}

= 2 α β \pm \sqrt{2 α^{2} β^{2} + α^{2} + β^{2}}

α^{2}, β^{2} ⩾ 0 \Rightarrow 2 α^{2} β^{2} + α^{2} + β^{2} ⩾ 0

Hence the roots are real .

Commented by Tinkutara last updated on 19/Aug/17

Thank you .