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Question Number 55198 by ajfour last updated on 19/Feb/19

$$x^5-4x^4+6x^3+8x-32=0$$$$Findatleastoneroot.$$

Commented by arvinddayama00@gmail.com last updated on 19/Feb/19

$$allvelueofx=?$$

Answered by rahul 19 last updated on 19/Feb/19

$$x=2!$$

Commented by rahul 19 last updated on 19/Feb/19

simply by hit & trial . ����

Answered by tanmay.chaudhury50@gmail.com last updated on 19/Feb/19

$$x^5+6x^3+8x=4x^4+32$$$$righthandside{x}^{evenpower}sorighthand$$$$sideisalways+ve$$$$nowlefthandsideshouldbe+ve$$$$soxcannotbelessthanzero$$$$sox\nless 0$$$$now$$$$(x^5+6x^3+8x)-(4x^4+32)=0$$$$f\left(x\right)=(x^5+6x^3+8x)-(4x^4+32)$$$$f\left(x\right)<0atx=0$$$$f\left(x\right)<0atx=1$$$$sonorootinbetween(0,1)$$$$$$$$nowcheckingbetween1and2$$$$f\left(2\right)=32+48+16-64-32=0$$$$sox=2isaroot..$$$$f\left(1\right)=1+6+8-4-32=-21$$$$nowgraduallyweincreasethevalueofx$$$$suchas1.1,1.2,...andobservechangeofsign$$$$f\left(x\right)=y=(x^5+6x^3+8x)-(4x^4+32)$$$$\frac{dy}{dx}=5x^4+18x^2+8-16x^3$$$$\frac{△y}{△x}\approx \frac{dy}{dx}$$$$△y=\frac{dy}{dx}\times △x=(5+18+8-16)\times 0.1=1.5$$$$sof(1.1)=f\left(1\right)+△y=-21+1.5=-19.5$$$$f(2.1)=f\left(2\right)+△y=0+1.5=1.5>0$$$$f\left(3\right)=3^5+6\times 3^3+8\times 3-4\times 3^4-32$$$$=3^4(3-4)+6\times 27-8$$$$=-81+162-8>0$$$$sonorootbetween2.1and3$$$$...$$$$f\left(4\right)=4^5+6\times 4^3+32-4\times 4^4-32>0$$$$sonorootbetween3and4$$$$now...$$$$f\left(x\right)=x^5+6x^3+8x-(4x^4+32)$$$$f\left(x\right)=g\left(x\right)-h\left(x\right)$$$$\frac{df}{dx}=\frac{dg}{dx}-\frac{dh}{dx}=(5x^4+18x^2+8)-\left(16x^3\right)$$$$=x^3(5x-16)+18x^2+8..$$$$nowlookwhenx>3.2\frac{df}{dx}>0$$$$\mathit{s}\mathit{o}\mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathit{w}\mathit{i}\mathit{l}\mathit{l}\mathit{b}\mathit{e}\mathit{n}\mathit{o}\mathit{s}\mathit{i}\mathit{g}\mathit{n}\mathit{c}\mathit{h}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{x}>3.2$$$$\mathit{s}\mathit{o}\mathit{t}\mathit{h}\mathit{i}\mathit{s}\mathit{e}\mathit{q}\mathit{n}\mathit{h}\mathit{a}\mathit{s}\mathit{o}\mathit{n}\mathit{l}\mathit{y}\mathit{o}\mathit{n}\mathit{e}\mathit{r}\mathit{o}\mathit{o}\mathit{t}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathit{i}\mathit{s}$$$$\mathit{x}=2...\mathit{p}\mathit{l}\mathit{s}\mathit{p}\mathit{a}\mathit{y}\mathit{y}\mathit{o}\mathit{u}\mathit{r}\mathit{k}\mathit{i}\mathit{n}\mathit{d}\mathit{a}\mathit{t}\mathit{t}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{o}\mathit{n}...$$$$$$$$$$$$$$

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