885/VB02. problem / Math - Solved Math Problems

MR-885 / 431c. problem

Solve the following equation:

$(1 + \sqrt{2}) x^{2} - 2 (1 - \sqrt{2}) x - 3 \sqrt{2} + 1 = 0$

Source: Vene T. Bogoslavov: Collection of Solved Problems #2 - 431.c.)

$(1 + \sqrt{2}) x^{2} - 2 (1 - \sqrt{2}) x - 3 \sqrt{2} + 1 = 0$

$a = 1 + \sqrt{2}, b = - 2 (1 - \sqrt{2}), c = - 3 \sqrt{2} + 1$

$x_{1, 2} = \frac{- (- 2 (1 - \sqrt{2})) \pm \sqrt{{(- 2 (1 - \sqrt{2}))}^{2} - 4 \cdot (1 + \sqrt{2}) \cdot (- 3 \sqrt{2} + 1)}}{2 \cdot (1 + \sqrt{2})}$

$x_{1, 2} = \frac{- (- 2 + 2 \sqrt{2}) \pm \sqrt{{(- 2 + 2 \sqrt{2})}^{2} - (4 \cdot 1 + 4 \cdot \sqrt{2}) \cdot (- 3 \sqrt{2} + 1)}}{2 \cdot 1 + 2 \cdot \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{{(- 2)}^{2} + 2 \cdot (- 2) \cdot (2 \sqrt{2}) + {(2 \sqrt{2})}^{2} - (4 + 4 \sqrt{2}) \cdot (- 3 \sqrt{2} + 1)}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{4 - 8 \sqrt{2} + 4 \cdot 2 - (4 \cdot (- 3 \sqrt{2}) + 4 \cdot 1 + 4 \sqrt{2} \cdot (- 3 \sqrt{2}) + 4 \sqrt{2} \cdot 1)}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{4 - 8 \sqrt{2} + 8 - (- 12 \sqrt{2} + 4 - 24 + 4 \sqrt{2})}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{4 - 8 \sqrt{2} + 8 + 12 \sqrt{2} - 4 + 24 - 4 \sqrt{2}}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{32}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{4 \cdot 4 \cdot 2}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm \sqrt{4} \cdot \sqrt{4} \cdot \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm 2 \cdot 2 \cdot \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{1, 2} = \frac{2 - 2 \sqrt{2} \pm 4 \cdot \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{1} = \frac{2 - 2 \sqrt{2} + 4 \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{1} = \frac{2 + 2 \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{1} = 1$

$x_{2} = \frac{2 - 2 \sqrt{2} - 4 \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{2} = \frac{2 - 6 \sqrt{2}}{2 + 2 \sqrt{2}}$

$x_{2} = \frac{2 - 6 \sqrt{2}}{2 + 2 \sqrt{2}} \cdot \frac{2 - 2 \sqrt{2}}{2 - 2 \sqrt{2}}$

$x_{2} = \frac{2 \cdot 2 + 2 \cdot (- 2 \sqrt{2}) + (- 6 \sqrt{2}) \cdot 2 + (- 6 \sqrt{2}) \cdot (- 2 \sqrt{2})}{2 \cdot 2 + 2 \cdot (- 2 \sqrt{2}) + 2 \sqrt{2} \cdot 2 + 2 \sqrt{2} \cdot (- 2 \sqrt{2})}$

$x_{2} = \frac{4 - 4 \sqrt{2} - 12 \sqrt{2} + 12 \cdot 2}{4 - 4 \sqrt{2} + 4 \sqrt{2} - 4 \cdot 2}$

$x_{2} = \frac{4 - 16 \sqrt{2} + 24}{4 - 8}$

$x_{2} = \frac{28 - 16 \sqrt{2}}{- 4}$

$x_{2} = \frac{28}{- 4} + \frac{- 16 \sqrt{2}}{- 4}$

$x_{2} = 4 \sqrt{2} - 7$

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