Question

Proszę o szybkie rozwiązanie ‼️ daje naj ‼️

Answer 1

Równanie 1.

[tex]x^4-13x^2+36=0\\\\t=x^2\\\\t^2-13t+36=0\\\\\Delta=(-13)^2-4*1*36=169-144=25\\\\\sqrt\Delta=5\\\\t_1=\frac{13-5}{2*1}=\frac{8}{2}=4\\\\t_2=\frac{13+5}{2*1}=\frac{18}{2}=9\\\\x^2=4\quad\vee\quad x^2=9\\\\x=2\quad\vee\quad x=-2\quad\vee\quad x=3\quad\vee\quad x=-3\\\\x\in\{-3,-2,2,3\}[/tex]

Równanie 2.

[tex]-x^4+7x^2-12=0\\\\t=x^2\\\\-t^2+7t-12=0\ |:(-1)\\\\t^2-7t+12=0\\\\\Delta=(-7)^2-4*1*12=49-48=1\\\\\sqrt\Delta=1\\\\t_1=\frac{7-1}{2*1}=\frac{6}{2}=3\\\\t_2=\frac{7+1}{2*1}=\frac{8}{2}=4\\\\x^2=3\quad\vee\quad x^2=4\\\\x=\sqrt3\quad\vee\quad x=-\sqrt3\quad\vee\quad x=2\quad\vee\quad x=-2\\\\x\in\{-2,-\sqrt3,\sqrt3,2\}[/tex]

Równanie 3.

[tex]9x^4+6x^2+1=0\\\\t=x^2\\\\9t^2+6t+1=0\\\\\Delta=6^2-4*9*1=36-36=0\\\\t_0=\frac{-6}{2*9}=\frac{-6}{18}=-\frac{1}{3} < 0\quad\text{odrzucamy}[/tex]

równanie sprzeczne

Równanie 4.

[tex]16x^4-1=8x^2\\\\16x^4-8x^2-1=0\\\\t=x^2\\\\16t^2-8t-1=0\\\\\Delta=(-8)^2-4*16*(-1)=64+64=128\\\\\sqrt\Delta=\sqrt{128}=\sqrt{64*2}=8\sqrt2\\\\t_1=\frac{8-8\sqrt2}{2*16}=\frac{8-8\sqrt2}{32}=\frac{1-\sqrt2}{4} < 0\quad\text{odrzucamy}\\\\t_2=\frac{8+8\sqrt2}{2*16}=\frac{8+8\sqrt2}{32}=\frac{1+\sqrt2}{4}\\\\x^2=\frac{1+\sqrt2}{4}\\\\x=\sqrt{\frac{1+\sqrt2}{4}}\quad\vee\quad x=-\sqrt{\frac{1+\sqrt2}{4}}\\\\x=\frac{\sqrt{1+\sqrt2}}{2}\quad\vee\quad x=-\frac{\sqrt{1+\sqrt2}}{2}[/tex]

[tex]x\in\left\{-\frac{\sqrt{1+\sqrt2}}{2},\frac{\sqrt{1+\sqrt2}}{2}\right\}[/tex]

Równanie 5.

[tex]12x^2+6=2-9x^4\\\\9x^4+12x^2+4=0\\\\t=x^2\\\\9t^2+12t+4=0\\\\\Delta=12^2-4*9*4=144-144=0\\\\t_0=\frac{-12}{2*9}=\frac{-12}{18}=-\frac{2}{3} < 0\quad\text{odrzucamy}[/tex]

równanie sprzeczne