Whakaoti i te 28r^3+7r=40r^2+10

$$28 r ^ { 3 } + 7 r = 40 r ^ { 2 } + 10$$

Whakaoti mō r

$$r = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

Ngā Upane Otinga

Tangohia te 40r^{2} mai i ngā taha e rua.

$$28r^{3}+7r-40r^{2}=10$$

Tangohia te 10 mai i ngā taha e rua.

$$28r^{3}+7r-40r^{2}-10=0$$

Hurinahatia te whārite ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$28r^{3}-40r^{2}+7r-10=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -10, ā, ka wehea e q te whakarea arahanga 28. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{5}{14},±\frac{5}{7},±\frac{10}{7},±\frac{5}{2},±5,±10,±\frac{5}{28},±\frac{5}{4},±\frac{1}{14},±\frac{1}{7},±\frac{2}{7},±\frac{1}{2},±1,±2,±\frac{1}{28},±\frac{1}{4}$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$r=\frac{10}{7}$$

Mā te whakatakotoranga Tauwehe, he tauwehe te r-k o te pūrau mō ia pūtake k. Whakawehea te 28r^{3}-40r^{2}+7r-10 ki te 7\left(r-\frac{10}{7}\right)=7r-10, kia riro ko 4r^{2}+1. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$4r^{2}+1=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 4 mō te a, te 0 mō te b, me te 1 mō te c i te ture pūrua.

$$r=\frac{0±\sqrt{0^{2}-4\times 4\times 1}}{2\times 4}$$

Mahia ngā tātaitai.

$$r=\frac{0±\sqrt{-16}}{8}$$

Tā te mea e kore te pūrua o tētahi tau tōraro e tautohutia ki te āpure tūturu, kāhore he rongoā.

$$r\in \emptyset$$

Rārangitia ngā otinga katoa i kitea.

$$r=\frac{10}{7}$$

Whakaoti mō r (complex solution)

$$r=-\frac{1}{2}i=-0.5i$$

$$r = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

$$r=\frac{1}{2}i=0.5i$$