Verified answer

Ответ:

[tex]3; \quad 7;[/tex]

Объяснение:

[tex](x-6)^{4}+(x-4)^{4}=82;[/tex]

[tex]((x-6)^{2})^{2}+((x-4)^{2})^{2}=82;[/tex]

[tex]((x-6)^{2})^{2}-2 \cdot (x-6)^{2} \cdot (x-4)^{2}+((x-4)^{2})^{2}+2 \cdot (x-6)^{2} \cdot (x-4)^{2}=82;[/tex]

[tex]((x-6)^{2}-(x-4)^{2})^{2}+2 \cdot (x-6)^{2} \cdot (x-4)^{2}=82;[/tex]

[tex](x^{2}-12x+36-(x^{2}-8x+16))^{2}+2 \cdot (x-6)^{2} \cdot (x-4)^{2}=82;[/tex]

[tex](x^{2}-12x+36-x^{2}+8x-16)^{2}+2 \cdot (x-6)^{2} \cdot (x-4)^{2}=82;[/tex]

[tex](20-4x)^{2}+2 \cdot (x-6)^{2} \cdot (x-4)^{2}=82;[/tex]

[tex](4x-20)^{2}+2 \cdot (x-6)^{2} \cdot (x-4)^{2}=82;[/tex]

[tex](4(x-5))^{2}+2 \cdot (x-5-1)^{2} \cdot (x-5+1)^{2}=82;[/tex]

[tex]4^{2}(x-5)^{2}+2 \cdot (x-5-1)^{2}\cdot (x-5+1)^{2}=82;[/tex]

[tex]16(x-5)^{2}+2(x-5-1)^{2}(x-5+1)^{2}=82;[/tex]

[tex]t=x-5;[/tex]

[tex]16t^{2}+2(t-1)^{2}(t+1)^{2}=82;[/tex]

[tex]16t^{2}+2((t-1)(t+1))^{2}=82;[/tex]

[tex]16t^{2}+2(t^{2}-1)^{2}=82 \quad |:2[/tex]

[tex]t^{4}-2t^{2}+1+8t^{2}-41=0;[/tex]

[tex]t^{4}+6t^{2}-40=0;[/tex]

[tex](t^{2})^{2}+6t^{2}-40=0;[/tex]

[tex]k=t^{2};[/tex]

[tex]k^{2}+6k-40=0;[/tex]

[tex]\displaystyle \left \{ {{k_{1}+k_{2}=-6} \atop {k_{1} \cdot k_{2}=-40}} \right. \Leftrightarrow \left \{ {{k_{1}=-10} \atop {k_{2}=4}} \right. ;[/tex]

Корень k₁ не имеет смысла.

[tex]t^{2}=4 \Rightarrow t=\pm\sqrt{4}=\pm 2;[/tex]

[tex]x-5=\pm 2 \Rightarrow x=5 \pm 2 \Rightarrow x=7 \ , \ x=3 \ ;[/tex]