Question

The product of the real roots of the equation $(x-1{)}^4+(x-5{)}^4=82$ is

• A. $-25$
• B. $8$
• C. $-8$
• D. $25$

Answer 1

The correct option is B $8$

Assuming
$y=\frac{(x-1)+(x-5)}{2}\Rightarrow x-3=y\Rightarrow x=y+3$

Given equation is
$(x-1{)}^4+(x-5{)}^4=82$
$(y+2{)}^4+(y-2{)}^4=82$
$\Rightarrow (y^2+4+4y{)}^2+(y^2+4-4y{)}^2=82$
$\Rightarrow 2\left[\right(y^2+4{)}^2+\left(4y{)}^2\right]=82\Rightarrow (y^2+4{)}^2+(4y{)}^2=41\Rightarrow y^4+24y^2-25=0\Rightarrow (y^2+25)(y^2-1)=0$
$\Rightarrow y^2-1=0(y^2+25\ne 0)\Rightarrow y=\pm 1\Rightarrow x-3=\pm 1\Rightarrow x=4,2$

Hence, the product of the real roots is $8.$