Question Number 62045 by hovea cw last updated on 14/Jun/19
$$\mathrm{help} \\ $$$$ \\ $$$$\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} +\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} =\mathrm{4} \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$
Answered by MJS last updated on 14/Jun/19
![(x+1)^(1/2) +(x^2 −1)^(1/3) =4 (x^2 −1)^(1/3) =4−(x+1)^(1/2) both sides^3 x^2 −1=12x+76−(x+49)(√(x+1)) 77+12x−x^2 =(x+49)(√(x+1)) both sides^2 [we will get some false solutions] x^4 −25x^3 −109x^2 −651x+3528=0 trying factors if 3528 ⇒ x_1 =3 looking at the graph of f(x)=(x+1)^(1/2) +(x^2 −1)^(1/3) −4 we see it′s the only real solution if you need complex solutions you must solve (x−3)(x^3 −22x^2 −175x−1176)=0 using Cardano′s formula](Q62060.png)
$$\left({x}+\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} =\mathrm{4} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} =\mathrm{4}−\left({x}+\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} \\ $$$$\mathrm{both}\:\mathrm{sides}\:^{\mathrm{3}} \\ $$$${x}^{\mathrm{2}} −\mathrm{1}=\mathrm{12}{x}+\mathrm{76}−\left({x}+\mathrm{49}\right)\sqrt{{x}+\mathrm{1}} \\ $$$$\mathrm{77}+\mathrm{12}{x}−{x}^{\mathrm{2}} =\left({x}+\mathrm{49}\right)\sqrt{{x}+\mathrm{1}} \\ $$$$\mathrm{both}\:\mathrm{sides}\:^{\mathrm{2}} \:\left[\mathrm{we}\:\mathrm{will}\:\mathrm{get}\:\mathrm{some}\:\mathrm{false}\:\mathrm{solutions}\right] \\ $$$${x}^{\mathrm{4}} −\mathrm{25}{x}^{\mathrm{3}} −\mathrm{109}{x}^{\mathrm{2}} −\mathrm{651}{x}+\mathrm{3528}=\mathrm{0} \\ $$$$\mathrm{trying}\:\mathrm{factors}\:\mathrm{if}\:\mathrm{3528}\:\Rightarrow\:{x}_{\mathrm{1}} =\mathrm{3} \\ $$$$\mathrm{looking}\:\mathrm{at}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of}\:{f}\left({x}\right)=\left({x}+\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} −\mathrm{4} \\ $$$$\mathrm{we}\:\mathrm{see}\:\mathrm{it}'\mathrm{s}\:\mathrm{the}\:\mathrm{only}\:\mathrm{real}\:\mathrm{solution} \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{need}\:\mathrm{complex}\:\mathrm{solutions}\:\mathrm{you}\:\mathrm{must}\:\mathrm{solve} \\ $$$$\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{3}} −\mathrm{22}{x}^{\mathrm{2}} −\mathrm{175}{x}−\mathrm{1176}\right)=\mathrm{0} \\ $$$$\mathrm{using}\:\mathrm{Cardano}'\mathrm{s}\:\mathrm{formula} \\ $$