solve-in-R-x-2-2x-5-2x-2-4x-8-7-

Question Number 167879 by mnjuly1970 last updated on 28/Mar/22

s o l v e i n R

\sqrt{x^{2} - 2 x + 5} + \sqrt{2 x^{2} - 4 x + 8} = 7

- - - - -

Commented by cortano1 last updated on 28/Mar/22

2 x^{2} - 4 x + 8 = 2 (x^{2} - 2 x + 4)

l e t x^{2} - 2 x + 4 = u

\sqrt{u + 1} + \sqrt{2 u} = 7

2 u = 49 + u + 1 - 14 \sqrt{u + 1}

u - 50 = - 14 \sqrt{u + 1}

196 u + 196 = u^{2} - 100 u + 250

u^{2} - 296 u + 250 = 0

{\begin{cases} u = 148 + 3 \sqrt{2406} \\ u = 148 - 3 \sqrt{2406} \end{cases}

{\begin{cases} x^{2} - 2 x + 4 = 148 + 3 \sqrt{2406} \\ x^{2} - 2 x + 4 = 148 - 3 \sqrt{2406} \end{cases}

Answered by MJS_new last updated on 28/Mar/22

\sqrt{x^{2} - 2 x + 5} + \sqrt{2 (x^{2} - 2 x + 4)} = 7

t = x - 1

\sqrt{t^{2} + 4} + \sqrt{2 (t^{2} + 3)} = 7

squaring (beware of false solutions!) & transforming

2 \sqrt{t^{2} + 4} \sqrt{2 (t^{2} + 3)} = 3 (13 - t^{2})

squaring (beware of false solutions!) & transforming

t^{4} - 290 t^{2} + 1425 = 0

(t^{2} - 285) (t^{2} - 5) = 0

t = \pm \sqrt{285} \lor t = \pm \sqrt{5}

testing all solutions \Rightarrow

t = \pm \sqrt{5}

\Rightarrow

x = 1 \pm \sqrt{5}

Commented by mnjuly1970 last updated on 28/Mar/22

b r a v o b r a v o m a s t e r \dots

Answered by nurtani last updated on 28/Mar/22

\sqrt{x^{2} - 2 x + 5} - \sqrt{2 x^{2} - 4 x + 8} = 7

\sqrt{x^{2} - 2 x + 5} - \sqrt{2 (x^{2} - 2 x + 4)} = 7

\sqrt{x^{2} - 2 x + 5} - \sqrt{2 (x^{2} - 2 x + 5 - 1)} = 7

x^{2} - 2 x + 5 = α

\sqrt{α} - \sqrt{2 (α - 1)} = 7

\sqrt{α} - \sqrt{2 α - 2} = 7

{(\sqrt{α} - \sqrt{2 α - 2})}^{2} = 7^{2}

α - 2 \sqrt{α} \cdot \sqrt{2 α - 2} + 2 α - 2 = 49

α - 2 \sqrt{α (2 α - 2)} + 2 α - 2 = 49

3 α - 2 \sqrt{2 α^{2} - 2 α} = 51

3 α - 51 = 2 \sqrt{2 α^{2} - 2 α}

{(3 α - 51)}^{2} = {(2 \sqrt{2 α^{2} - 2 α})}^{2}

9 α^{2} - 306 α + 2601 = 4 (2 α^{2} - 2 α)

9 α^{2} - 306 α + 2601 = 8 α^{2} - 8 α

α^{2} - 298 α + 2601 = 0

(α - 289) (α - 9) = 0 {\begin{cases} α = 289 \\ α = 9 \end{cases}

{\begin{cases} x^{2} - 2 x + 5 = 289 \Rightarrow x^{2} - 2 x - 284 = 0 {\begin{cases} x_{1} = \sqrt{285} + 1 \\ x_{2} = - \sqrt{285} + 1 \end{cases} \\ x^{2} - 2 x + 5 = 9 \Rightarrow x^{2} - 2 x - 4 = 0 {\begin{cases} x_{3} = \sqrt{5} + 1 \\ x_{4} = - \sqrt{5} + 1 \end{cases} \end{cases}

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