Whakaoti i te 2x^4-7x^3-7x^2+35x-15=0

$$2 x ^ { 4 } - 7 x ^ { 3 } - 7 x ^ { 2 } + 35 x - 15 = 0$$

Whakaoti mō x

$$x=\sqrt{5}\approx 2.236067977$$

$$x=-\sqrt{5}\approx -2.236067977$$

$$x=3$$

$$x=\frac{1}{2}=0.5$$

Ngā Upane Otinga

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 -15, ā, ka wehea e q te whakarea arahanga 2. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{15}{2},±15,±\frac{5}{2},±5,±\frac{3}{2},±3,±\frac{1}{2},±1$$

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.

$$x=3$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 2x^{4}-7x^{3}-7x^{2}+35x-15 ki te x-3, kia riro ko 2x^{3}-x^{2}-10x+5. Whakaotihia te whārite ina ōrite te hua ki te 0.

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

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 5, ā, ka wehea e q te whakarea arahanga 2. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{5}{2},±5,±\frac{1}{2},±1$$

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 2x^{3}-x^{2}-10x+5 ki te 2\left(x-\frac{1}{2}\right)=2x-1, kia riro ko x^{2}-5. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{2}-5=0$$

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

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-5\right)}}{2}$$

Mahia ngā tātaitai.

$$x=\frac{0±2\sqrt{5}}{2}$$

Whakaotia te whārite x^{2}-5=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=-\sqrt{5} x=\sqrt{5}$$

Rārangitia ngā otinga katoa i kitea.

$$x=3 x=\frac{1}{2} x=-\sqrt{5} x=\sqrt{5}$$