The product of the real roots of the equation | vedclass

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(C) Given equation: $(x+1)^4+(x+3)^4=8$. Let $x+2=y$,then $x+1=y-1$ and $x+3=y+1$. The equation becomes $(y-1)^4+(y+1)^4=8$. Expanding using binomial

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$7-2 \sqrt{3}$

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(C) Given equation: $(x+1)^4+(x+3)^4=8$. Let $x+2=y$,then $x+1=y-1$ and $x+3=y+1$. The equation becomes $(y-1)^4+(y+1)^4=8$. Expanding using binomial theorem: $(y^4-4y^3+6y^2-4y+1)+(y^4+4y^3+6y^2+4y+1)=8$. $2y^4+12y^2+2=8$ $\Rightarrow 2y^4+12y^2-6=0$ $\Rightarrow y^4+6y^2-3=0$. Let $t=y^2$,then $t^2+6t-3=0$. $t = \frac{-6 \pm \sqrt{36-4(1)(-3)}}{2} = \frac{-6 \pm \sqrt{48}}{2} = -3 \pm 2\sqrt{3}$. Since $t=y^2$ must be non-negative for real $y$,we take $y^2 = 2\sqrt{3}-3$. Substituting back $y=x+2$: $(x+2)^2 = 2\sqrt{3}-3$. $x^2+4x+4 = 2\sqrt{3}-3 \Rightarrow x^2+4x+(7-2\sqrt{3}) = 0$. This is a quadratic equation in $x$ of the form $ax^2+bx+c=0$. The product of the roots is $\frac{c}{a} = \frac{7-2\sqrt{3}}{1} = 7-2\sqrt{3}$.

The product of the real roots of the equation $(x+1)^4+(x+3)^4=8$ is

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