Whakaoti i te 5x^2-2x-7=0

$$5 x ^ { 2 } - 2 x - 7 = 0$$

Whakaoti mō x

$$x=-1$$

$$x = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=5\left(-7\right)=-35$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -35.

$$1,-35 5,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-35=-34 5-7=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-7 b=5$$

Tuhia anō te 5x^{2}-2x-7 hei \left(5x^{2}-7x\right)+\left(5x-7\right).

$$\left(5x^{2}-7x\right)+\left(5x-7\right)$$

Whakatauwehea atu x i te 5x^{2}-7x.

$$x\left(5x-7\right)+5x-7$$

Whakatauwehea atu te kīanga pātahi 5x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-7\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 5x-7=0 me te x+1=0.

$$x=\frac{7}{5} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$5x^{2}-2x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -2 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 5\left(-7\right)}}{2\times 5}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 5\left(-7\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-2\right)±\sqrt{4-20\left(-7\right)}}{2\times 5}$$

Whakareatia -20 ki te -7.

$$x=\frac{-\left(-2\right)±\sqrt{4+140}}{2\times 5}$$

Tāpiri 4 ki te 140.

$$x=\frac{-\left(-2\right)±\sqrt{144}}{2\times 5}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-2\right)±12}{2\times 5}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±12}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{2±12}{10}$$

Nā, me whakaoti te whārite x=\frac{2±12}{10} ina he tāpiri te ±. Tāpiri 2 ki te 12.

$$x=\frac{14}{10}$$

Whakahekea te hautanga \frac{14}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{7}{5}$$

Nā, me whakaoti te whārite x=\frac{2±12}{10} ina he tango te ±. Tango 12 mai i 2.

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

Whakawehe -10 ki te 10.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{7}{5} x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$5x^{2}-2x-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$5x^{2}-2x-7-\left(-7\right)=-\left(-7\right)$$

Mā te tango i te -7 i a ia ake anō ka toe ko te 0.

$$5x^{2}-2x=-\left(-7\right)$$

Tango -7 mai i 0.

$$5x^{2}-2x=7$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-2x}{5}=\frac{7}{5}$$

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

$$x^{2}-\frac{2}{5}x=\frac{7}{5}$$

Whakawehea te -\frac{2}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{5}. Nā, tāpiria te pūrua o te -\frac{1}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{7}{5}+\left(-\frac{1}{5}\right)^{2}$$

Pūruatia -\frac{1}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{7}{5}+\frac{1}{25}$$

Tāpiri \frac{7}{5} ki te \frac{1}{25} 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.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{36}{25}$$

Tauwehea x^{2}-\frac{2}{5}x+\frac{1}{25}. 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}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{36}{25}$$

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

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{36}{25}}$$

Whakarūnātia.

$$x-\frac{1}{5}=\frac{6}{5} x-\frac{1}{5}=-\frac{6}{5}$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$x=\frac{7}{5} x=-1$$