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Question Number 126887 by mnjuly1970 last updated on 25/Dec/20

$$...calculus...$$$$pleasesolve:\left(withexplanation\right)$$$$\{{}_{x^2+y=3}^{x+y^2=5}$$$$$$

Commented by Lordose last updated on 25/Dec/20

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{a}\mathrm{calculus}\mathrm{approach}\mathrm{though}$$

Commented by bramlexs22 last updated on 25/Dec/20

$$x=5-y^2$$$${(5-y^2)}^2+y=3$$$$y^4-10y^2+y+22=0$$$$(y-2)(y^3+2y^2-6y-11)=0$$$$y=2\wedge x=1$$$$$$$$y^3+2y^2-6y-11=0$$$$Cardano$$

Answered by Lordose last updated on 25/Dec/20

$$$$$$\mathrm{x}+{\mathrm{y}}^2=5---\left(1\right)$$$${\mathrm{x}}^2+\mathrm{y}=3---\left(2\right)$$$$\mathrm{From}\left(1\right),\mathrm{x}=5-{\mathrm{y}}^2$$$${(5-{\mathrm{y}}^2)}^2+\mathrm{y}=3$$$$25-{10\mathrm{y}}^2+{\mathrm{y}}^4+\mathrm{y}=3$$$${\mathrm{y}}^4-{10\mathrm{y}}^2+\mathrm{y}+22=0$$$$(\mathrm{y}-2)({\mathrm{y}}^3+{2\mathrm{y}}^2-6\mathrm{x}-11)=0$$$$\mathrm{Only}\mathrm{Integer}\mathrm{value}\mathrm{for}\mathrm{y}=2$$$$\mathrm{sub}\mathrm{in}\left(1\right)$$$$\mathrm{x}=5-2^2=1$$$$(\mathrm{x},\mathrm{y})=(1,2)$$

Answered by mr W last updated on 25/Dec/20

$$x+{(3-x^2)}^2=5$$$$x^4-6x^2+x+4=0$$$$(x-1)(x^3+x^2-5x-4)=0$$$$\Rightarrow x-1=0\Rightarrow x=1$$$$x^3+x^2-5x-4=0$$$${(t-\frac{1}{3})}^3+{(t-\frac{1}{3})}^2-5(t-\frac{1}{3})-4=0$$$$t^3-t^2+\frac{t}{3}-\frac{1}{27}+t^2-\frac{2t}{3}+\frac{1}{9}-5t+\frac{5}{3}-4=0$$$$t^3-\frac{16t}{3}-\frac{61}{27}=0$$$$t=2\sqrt{\frac{16}{9}}\sin (\frac{1}{3}{\sin }^{-1}\frac{-\frac{61}{54}}{\frac{16}{9}\sqrt{\frac{16}{9}}}+\frac{2k\pi }{3})$$$$t=\frac{8}{3}\sin (-\frac{1}{3}{\sin }^{-1}\frac{61}{128}+\frac{2k\pi }{3})(k=0,1,2)$$$$\Rightarrow x=\frac{8}{3}\sin (-\frac{1}{3}{\sin }^{-1}\frac{61}{128}+\frac{2k\pi }{3})-\frac{1}{3}(k=0,1,2)$$$$\approx -0.7729,2.1642,-2.3914$$

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