Rovnice vyšších stupňů

1. V množině reálných čísel řešte rovnice

$$(a)\ x^4-6x^2+5=0$$ $$(b)\ x^4-5x^2+6=0$$

$$(c)\ x^4-5x^2-24=0$$ $$(d)\ x^4-10x^2+9=0$$

$$(e)\ x^4-12x^2-64=0$$ $$(f)\ x^4-16x^2-80=0$$

$$(g)\ x^4-3(x^2-1)=7(x^2-3)$$ $$(h)\ 4x^4-37x^2+9=0$$

$$(i)\ 8x^4-6x^2+1=0$$ $$(j)\ 9x^4=37x^2-4$$

$$(k)\ x^6-2x^3-3=0$$ $$(l)\ x^6-28x^3+27=0$$

$$a)\ x\in\left\{\pm1; \pm \sqrt5\right\}$$ $$b)\ x\in\left\{\pm\sqrt2; \pm\sqrt3\right\}$$

$$c)\ x\in\left\{\pm 2\sqrt2\right\}$$ $$d)\ x\in\left\{\pm1; \pm3\right\}$$

$$e)\ x\in\left\{\pm4\right\}$$ $$f)\ x\in\left\{\pm 2\sqrt5\right\}$$

$$g)\ x\in\left\{\pm2; \pm\sqrt6\right\}$$ $$h)\ x\in\left\{\pm3;\pm \frac12\right\}$$

$$i)\ x\in\left\{\pm \frac12; \pm \frac{\sqrt2}{2}\right\}$$ $$j)\ x\in\left\{\pm2; \pm\frac{1}{3}\right\}$$

$$k)\ x\in\left\{-1; \sqrt[3]{3}\right\}$$ $$l)\ x\in\left\{1;3\right\}$$

2. V množině reálných čísel řešte rovnice

$$(a)\ (x^2-4x-5)(x-3)=0$$ $$(b)\ (x+1)(3x^2-3x+2)=0$$

$$(c)\ 5x^6-20x^4=0$$ $$(d)\ x^4-8x=0$$

$$(f)\ x^3=3x^2$$ $$(f)\ x^3-4x^2+4x=0$$

$$(g)\ x^3-6x=x^2$$ $$(h)\ x^4-6x^2+9=3x^2-9$$

$$a)\ x\in\left\{-1; 3; 5\right\}$$ $$b)\ x\in\left\{-1\right\}$$

$$c)\ x\in\left\{0; \pm 2\right\}$$ $$d)\ x\in\left\{0; 2\right\}$$

$$e)\ x\in\left\{0; 3\right\}$$ $$f)\ x\in\left\{0; 2\right\}$$

$$g)\ x\in\left\{-2; 0; 3\right\}$$ $$h)\ x\in\left\{\pm \sqrt3;\pm \sqrt6\right\}$$

$$(a)\ x^4-6x^3-2x^2+6x+1=0$$ $$(b)\ 4x^3-8x^2+5x-1=0$$

$$(c)\ x^4-4x^3+3x^2+4x-4=0$$ $$(d)\ x^3-6x^2+11x-6=0$$

$$a)\ x\in\left\{\pm 1; 3\pm \sqrt{10}\right\}$$ $$b)\ x\in\left\{\frac12; 1\right\}$$

$$c)\ x\in\left\{\pm 1; 2\right\}$$ $$d)\ x\in\left\{1; 2; 3\right\}$$

4. V množině reálných čísel řešte rovnice

$$(a)\ (x^2-5x+5)^2-1=0$$ $$(b)\ (x^2-5x-1)^2-25=0$$

$$(c)\ x^4+6x^2+9=4x^3+12x$$ $$(d)\ x^4+6x^2+9=36x^2-72x+36$$

$$(e)\ (x^2+4x-2)^2=(5x^2-1)^2$$ $$(f)\ \frac{x^2-3}{2}=\frac{2}{x^2-3}$$

$$(g)\ (x^2-5x+2)^2+6(x^2-5x+1)+14=0$$ $$(h)\ (x-5)(x-3)(x-1)(x+1)=(x-1)(x+1)(x+3)(x+5)$$

$$a)\ x\in\left\{1; 2; 3; 4\right\}$$ $$b)\ x\in\left\{ -1; 1; 4; 6\right\}$$

$$c)\ x\in\left\{1; 3\right\}$$ $$d)\ x\in\left\{3; -3 \pm 2\sqrt3\right\}$$

$$e)\ x\in\left\{\frac12; \frac{-2 \pm \sqrt{22} }6 \right\}$$ $$f)\ x\in\left\{\pm1; \pm \sqrt5\right\}$$

$$g)\ x\in\left\{1; 2; 3; 4\right\}$$ $$h)\ x\in\left\{0; \pm 1\right\}$$

$$(a)\ x^4-6x^2+5=0$$	$$(b)\ x^4-5x^2+6=0$$
$$(c)\ x^4-5x^2-24=0$$	$$(d)\ x^4-10x^2+9=0$$
$$(e)\ x^4-12x^2-64=0$$	$$(f)\ x^4-16x^2-80=0$$
$$(g)\ x^4-3(x^2-1)=7(x^2-3)$$	$$(h)\ 4x^4-37x^2+9=0$$
$$(i)\ 8x^4-6x^2+1=0$$	$$(j)\ 9x^4=37x^2-4$$
$$(k)\ x^6-2x^3-3=0$$	$$(l)\ x^6-28x^3+27=0$$

$$a)\ x\in\left\{\pm1; \pm \sqrt5\right\}$$	$$b)\ x\in\left\{\pm\sqrt2; \pm\sqrt3\right\}$$
$$c)\ x\in\left\{\pm 2\sqrt2\right\}$$	$$d)\ x\in\left\{\pm1; \pm3\right\}$$
$$e)\ x\in\left\{\pm4\right\}$$	$$f)\ x\in\left\{\pm 2\sqrt5\right\}$$
$$g)\ x\in\left\{\pm2; \pm\sqrt6\right\}$$	$$h)\ x\in\left\{\pm3;\pm \frac12\right\}$$
$$i)\ x\in\left\{\pm \frac12; \pm \frac{\sqrt2}{2}\right\}$$	$$j)\ x\in\left\{\pm2; \pm\frac{1}{3}\right\}$$
$$k)\ x\in\left\{-1; \sqrt[3]{3}\right\}$$	$$l)\ x\in\left\{1;3\right\}$$

$$(a)\ (x^2-4x-5)(x-3)=0$$	$$(b)\ (x+1)(3x^2-3x+2)=0$$
$$(c)\ 5x^6-20x^4=0$$	$$(d)\ x^4-8x=0$$
$$(f)\ x^3=3x^2$$	$$(f)\ x^3-4x^2+4x=0$$
$$(g)\ x^3-6x=x^2$$	$$(h)\ x^4-6x^2+9=3x^2-9$$

$$a)\ x\in\left\{-1; 3; 5\right\}$$	$$b)\ x\in\left\{-1\right\}$$
$$c)\ x\in\left\{0; \pm 2\right\}$$	$$d)\ x\in\left\{0; 2\right\}$$
$$e)\ x\in\left\{0; 3\right\}$$	$$f)\ x\in\left\{0; 2\right\}$$
$$g)\ x\in\left\{-2; 0; 3\right\}$$	$$h)\ x\in\left\{\pm \sqrt3;\pm \sqrt6\right\}$$

$$(a)\ x^4-6x^3-2x^2+6x+1=0$$	$$(b)\ 4x^3-8x^2+5x-1=0$$
$$(c)\ x^4-4x^3+3x^2+4x-4=0$$	$$(d)\ x^3-6x^2+11x-6=0$$

$$a)\ x\in\left\{\pm 1; 3\pm \sqrt{10}\right\}$$	$$b)\ x\in\left\{\frac12; 1\right\}$$
$$c)\ x\in\left\{\pm 1; 2\right\}$$	$$d)\ x\in\left\{1; 2; 3\right\}$$

$$(a)\ (x^2-5x+5)^2-1=0$$	$$(b)\ (x^2-5x-1)^2-25=0$$
$$(c)\ x^4+6x^2+9=4x^3+12x$$	$$(d)\ x^4+6x^2+9=36x^2-72x+36$$
$$(e)\ (x^2+4x-2)^2=(5x^2-1)^2$$	$$(f)\ \frac{x^2-3}{2}=\frac{2}{x^2-3}$$
$$(g)\ (x^2-5x+2)^2+6(x^2-5x+1)+14=0$$	$$(h)\ (x-5)(x-3)(x-1)(x+1)=(x-1)(x+1)(x+3)(x+5)$$

$$a)\ x\in\left\{1; 2; 3; 4\right\}$$	$$b)\ x\in\left\{ -1; 1; 4; 6\right\}$$
$$c)\ x\in\left\{1; 3\right\}$$	$$d)\ x\in\left\{3; -3 \pm 2\sqrt3\right\}$$
$$e)\ x\in\left\{\frac12; \frac{-2 \pm \sqrt{22} }6 \right\}$$	$$f)\ x\in\left\{\pm1; \pm \sqrt5\right\}$$
$$g)\ x\in\left\{1; 2; 3; 4\right\}$$	$$h)\ x\in\left\{0; \pm 1\right\}$$