Solve-1-x-2-4-x-x-2-6x-11-2-x-4-x-3-2ax-2-ax-a-2-0-

Question Number 70898 by Henri Boucatchou last updated on 09/Oct/19

S o l v e :

1 .) \sqrt{x - 2} + \sqrt{4 - x} = x^{2} - 6 x + 11

2 .) x^{4} + x^{3} - 2 {a x}^{2} - a x + a^{2} = 0

Answered by MJS last updated on 09/Oct/19

2)

x^{4} + x^{3} - 2 {a x}^{2} - a x + a^{2} = 0

(x^{2} - a) (x^{2} + x - a) = 0

\Rightarrow x = \pm \sqrt{a} \lor x = - \frac{1}{2} \pm \frac{\sqrt{4 a + 1}}{2}

Answered by MJS last updated on 09/Oct/19

1)

\sqrt{x - 2} + \sqrt{4 - x} = x^{2} - 6 x + 11

t = x - 3 \Leftrightarrow x = t + 3

\sqrt{1 - t} + \sqrt{1 + t} = t^{2} + 2

squaring

2 + 2 \sqrt{t^{2} - 1} = t^{4} + 4 t^{2} + 4

2 \sqrt{1 - t^{2}} = t^{4} + 4 t^{2} + 2

squaring

4 - 4 t^{2} = 8 t^{6} + 20 t^{4} + 16 t^{2} + 4

t^{2} (t^{6} + 8 t^{4} + 20 t^{2} + 20) = 0

t_{1} = t_{2} = 0

no other real solution

\Rightarrow x = 3

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