# Differentation: x intercept

#### nikconsult

i have the problem how to calculate the x intercept for the attach graph. please refer graph.jpg

here i also attach my calculation how to get x intercept. it is very different from the solution.

hope some one con comment on it

tq(Crying)

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#### zzzoak

$$\displaystyle x^2-x^3+8=0$$- this is cubic equation.
It is solved more complicated than quadratic equation.

nikconsult

#### dwsmith

MHF Hall of Honor
i have the problem how to calculate the x intercept for the attach graph. please refer graph.jpg

here i also attach my calculation how to get x intercept. it is very different from the solution.

hope some one con comment on it

tq(Crying)
$$\displaystyle y=\frac{x^2(1-x)+8}{1-x}\to 0=x^2(1-x)+8\to -x^3+x^2+8=0$$

You can use Des Cartes Rule of Sign.

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#### nikconsult

$$\displaystyle y=\frac{x^2(1-x)+8}{1-x}\to 0=x^2(1-x)+8\to -x^3+x^2+8=0$$

You can use Des Cartes Rule of Sign and synthetic division to solve.

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why not we just factorize
$$\displaystyle -x^3+x^2+8=0$$

#### dwsmith

MHF Hall of Honor
why not we just factorize
$$\displaystyle -x^3+x^2+8=0$$
Ok what does it factor to?

#### nikconsult

Ok what does it factor to?

according to my attached calculation it is 9.but according to attached graph, it not exactly root is 2

#### dwsmith

MHF Hall of Honor
$$\displaystyle a=-1, b=1, c=0,$$ and $$\displaystyle d=8$$

$$\displaystyle x=\sqrt[3]{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)+\sqrt{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)^2+\left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}$$$$\displaystyle +\sqrt[3]{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)-\sqrt{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)^2+\left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}$$$$\displaystyle -\frac{b}{3a}$$

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#### nikconsult

$$\displaystyle a=-1, b=1, c=0,$$ and $$\displaystyle d=8$$

$$\displaystyle x=\sqrt[3]{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)+\sqrt{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)^2+\left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^2}}$$$$\displaystyle +\sqrt[3]{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)+\sqrt{\left(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\right)^2+\left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^2}}$$$$\displaystyle -\frac{b}{3a}$$
tq for the formulae ..so what is name of the formulae, i try to search it. is it Des Cartes Rule of Sign??

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but according to attached graph, it not exactly root is 2

ok i think i need to key in the a=-1, b=1, c=0 & d=8, into the above equation??

#### dwsmith

MHF Hall of Honor
(+) 1
(-) 0
(i) 2

but according to attached graph, it not exactly root is 2
That says there is 1 positive solution and 2 imaginary.

Formula Cardano.