Whakaoti i te 5x^2=11x-2

$$5 { x }^{ 2 } = 11x-2$$

Whakaoti mō x

$$x=\frac{1}{5}=0.2$$

$$x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 11x mai i ngā taha e rua.

$$5x^{2}-11x=-2$$

Me tāpiri te 2 ki ngā taha e rua.

$$5x^{2}-11x+2=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=5\times 2=10$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 10.

$$-1,-10 -2,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-10=-11 -2-5=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-10 b=-1$$

Tuhia anō te 5x^{2}-11x+2 hei \left(5x^{2}-10x\right)+\left(-x+2\right).

$$\left(5x^{2}-10x\right)+\left(-x+2\right)$$

Tauwehea te 5x i te tuatahi me te -1 i te rōpū tuarua.

$$5x\left(x-2\right)-\left(x-2\right)$$

Whakatauwehea atu te kīanga pātahi x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-2\right)\left(5x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-2=0 me te 5x-1=0.

$$x=2 x=\frac{1}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 11x mai i ngā taha e rua.

$$5x^{2}-11x=-2$$

Me tāpiri te 2 ki ngā taha e rua.

$$5x^{2}-11x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -11 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 5\times 2}}{2\times 5}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 5\times 2}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-11\right)±\sqrt{121-20\times 2}}{2\times 5}$$

Whakareatia -20 ki te 2.

$$x=\frac{-\left(-11\right)±\sqrt{121-40}}{2\times 5}$$

Tāpiri 121 ki te -40.

$$x=\frac{-\left(-11\right)±\sqrt{81}}{2\times 5}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-11\right)±9}{2\times 5}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±9}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{11±9}{10}$$

Nā, me whakaoti te whārite x=\frac{11±9}{10} ina he tāpiri te ±. Tāpiri 11 ki te 9.

$$x=\frac{20}{10}$$

Whakawehe 20 ki te 10.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{11±9}{10} ina he tango te ±. Tango 9 mai i 11.

$$x=\frac{2}{10}$$

Whakahekea te hautanga \frac{2}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{5}$$

Kua oti te whārite te whakatau.

$$x=2 x=\frac{1}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 11x mai i ngā taha e rua.

$$5x^{2}-11x=-2$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-11x}{5}=-\frac{2}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{11}{5}x=-\frac{2}{5}$$

Whakawehea te -\frac{11}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{10}. Nā, tāpiria te pūrua o te -\frac{11}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{5}x+\left(-\frac{11}{10}\right)^{2}=-\frac{2}{5}+\left(-\frac{11}{10}\right)^{2}$$

Pūruatia -\frac{11}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{5}x+\frac{121}{100}=-\frac{2}{5}+\frac{121}{100}$$

Tāpiri -\frac{2}{5} ki te \frac{121}{100} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{11}{5}x+\frac{121}{100}=\frac{81}{100}$$

Tauwehea x^{2}-\frac{11}{5}x+\frac{121}{100}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{10}\right)^{2}=\frac{81}{100}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{11}{10}\right)^{2}}=\sqrt{\frac{81}{100}}$$

Whakarūnātia.

$$x-\frac{11}{10}=\frac{9}{10} x-\frac{11}{10}=-\frac{9}{10}$$

Me tāpiri \frac{11}{10} ki ngā taha e rua o te whārite.

$$x=2 x=\frac{1}{5}$$