Whakaoti i te 5x^2+2x=7

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

Whakaoti mō x

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

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 7 mai i ngā taha e rua.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx-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 tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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.

$$a=-5 b=7$$

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

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

Tauwehea te 5x i te tuatahi me te 7 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

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

Me tango 7 mai i ngā taha e rua o te whārite.

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

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

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

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}.

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

Pūrua 2.

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

Whakareatia -4 ki te 5.

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

Whakareatia -20 ki te -7.

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

Tāpiri 4 ki te 140.

$$x=\frac{-2±\sqrt{144}}{2\times 5}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-2±12}{2\times 5}$$

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.

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

Whakawehe 10 ki te 10.

$$x=1$$

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

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

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

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

Kua oti te whārite te whakatau.

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

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.

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

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 tango \frac{1}{5} mai i ngā taha e rua o te whārite.

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