Question

Rozwiąż (x+2)(x+4)(x+6)(x+8)=81

Answer 1

Verified answer

Odpowiedź:

[tex]\displaystyle (x+2)(x+8)(x+4)(x+6)=81\\(x^2+10x+16)(x^2+10x+24)=81\\x^2+10x=t\\(t+16)(t+24)=81\\t^2+24t+16t+384=81\\t^2+40t+303=0\\\Delta=40^2-4\cdot303=388\\\sqrt{\Delta}=2\sqrt{97} \\t_1=\frac{-40+2\sqrt{97} }{2} =-20+\sqrt{97} \\t_2=-20-\sqrt{97}[/tex]

[tex]\displaystyle x^{2} +10x=-20+\sqrt{97} \\x^{2} +10x+20-\sqrt{97} =0\\\Delta=100-4(20-\sqrt{97} )=100-80+4\sqrt{97} =20+4\sqrt{97} \\\sqrt{\Delta}= 2\sqrt{5+\sqrt{97} } \\x_1=\frac{-10+2\sqrt{5+\sqrt{97} } }{2} =-5+\sqrt{5+\sqrt{97} }\\x_2=-5-\sqrt{5+\sqrt{97}[/tex]

[tex]\displaystyle x^{2} +10x=-20-\sqrt{97} \\x^{2} +10x+20+\sqrt{97} =0\\\Delta=100-4(20+\sqrt{97} )=100-80-4\sqrt{97} =20-4\sqrt{97} < 0[/tex]

[tex]\displaystyle x_1=-5+\sqrt{5+\sqrt{97} }\\x_2=-5-\sqrt{5+\sqrt{97}[/tex]