Question

dau coroana
$( \frac{ {x}^{2} + 6 } {x} ) {}^{2} = 25$

Answer 1

$\left(\dfrac{x^2+6}{x}\right)^2 = 25\\ \Leftrightarrow \, \left(\dfrac{x^2+6}{x}\right)^2 = 5^2 \,\Leftrightarrow \, \left(\dfrac{x^2+6}{x}\right)^2 = 5^2\\ \\\Leftrightarrow \, \dfrac{x^2+6}{x} = \pm 5 \,\Leftrightarrow \, x^2+6 = \pm 5x\\ \\ \Leftrightarrow \, x^2\mp 5x +6 = 0\\ \\ \Delta = (\mp 5)^2 - 4\cdot 1\cdot 6 = 25-24 = 1$

$\Rightarrow\,x_{1,2} = \dfrac{\pm 5 \pm \sqrt{1}}{2}\Rightarrow\, S = \left\{\dfrac{-5-1}{2};\,\dfrac{-5+1}{2};\,\dfrac{5-1}{2};\,\dfrac{5+1}{2}\right\}\\ \\ \Rightarrow \,\boxed{\boxed{S = \left\{-3;\,-2;\,2;\,3\right\}}}$

Answer 2

Răspuns:

Explicație pas cu pas:

[ ( x²+6)/x]² = 25

[ ( x²+6)/x]² = (±5)²

(x²+6)/x = +5

x²+6 = 5x

x² - 5x + 6 = 0

Δ = 25 - 24 ⇒  Δ = 1

x = ( 5 ±√Δ)/2 = (5±1)/2 =>   x₁ = 6/2 ⇒ x₁ = 3

                                             x₂ = 4/2 ⇒ x₂ = 2

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x = ( - 5±1)/2  ⇒   x = - 4/2 ⇒ x = - 2

                            x = - 6/2 ⇒ x = - 3

Solutii:  { - 3;  - 2;  2;  3}