Whakaoti i te =20x^4+31x^2-9

$$= 20 x ^ { 4 } + 31 x ^ { 2 } - 9$$

Tauwehe

$$\left(2x-1\right)\left(2x+1\right)\left(5x^{2}+9\right)$$

Ngā Upane Otinga

Kia tauwehea ai te kīanga, me whakaoti te whārite ina ōrite ki te 0.

$$20x^{4}+31x^{2}-9=0$$

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 -9, ā, ka wehea e q te whakarea arahanga 20. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{9}{20},±\frac{9}{10},±\frac{9}{5},±\frac{9}{4},±\frac{9}{2},±9,±\frac{3}{20},±\frac{3}{10},±\frac{3}{5},±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{20},±\frac{1}{10},±\frac{1}{5},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=\frac{1}{2}$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 20x^{4}+31x^{2}-9 ki te 2\left(x-\frac{1}{2}\right)=2x-1, kia riro ko 10x^{3}+5x^{2}+18x+9. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$10x^{3}+5x^{2}+18x+9=0$$

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 9, ā, ka wehea e q te whakarea arahanga 10. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{9}{10},±\frac{9}{5},±\frac{9}{2},±9,±\frac{3}{10},±\frac{3}{5},±\frac{3}{2},±3,±\frac{1}{10},±\frac{1}{5},±\frac{1}{2},±1$$

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.

$$x=-\frac{1}{2}$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 10x^{3}+5x^{2}+18x+9 ki te 2\left(x+\frac{1}{2}\right)=2x+1, kia riro ko 5x^{2}+9. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$5x^{2}+9=0$$

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 5 mō te a, te 0 mō te b, me te 9 mō te c i te ture pūrua.

$$x=\frac{0±\sqrt{0^{2}-4\times 5\times 9}}{2\times 5}$$

Mahia ngā tātaitai.

$$x=\frac{0±\sqrt{-180}}{10}$$

Kāore te pūrau 5x^{2}+9 i whakatauwehea i te mea kāhore ōna pūtake whakahau.

$$5x^{2}+9$$

Me tuhi anō te kīanga whakatauwehe mā ngā pūtake i riro.

$$\left(2x-1\right)\left(2x+1\right)\left(5x^{2}+9\right)$$

Aromātai

$$20x^{4}+31x^{2}-9$$