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

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

Whakaoti mō x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=-1$$

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 2x^{2}+ax+bx+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=2\times 5=10$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga 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 7.

$$a=2 b=5$$

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

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

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

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

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

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

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

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

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.

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

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

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

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 2\times 5}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-7±\sqrt{49-8\times 5}}{2\times 2}$$

Whakareatia -8 ki te 5.

$$x=\frac{-7±\sqrt{49-40}}{2\times 2}$$

Tāpiri 49 ki te -40.

$$x=\frac{-7±\sqrt{9}}{2\times 2}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-7±3}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-7±3}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±3}{4} ina he tāpiri te ±. Tāpiri -7 ki te 3.

$$x=-\frac{4}{4}$$

Whakawehe -4 ki te 4.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-7±3}{4} ina he tango te ±. Tango 3 mai i -7.

$$x=-\frac{10}{4}$$

Whakahekea te hautanga \frac{-10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

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.

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

Me tango 5 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te 2.

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

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

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

Whakawehea te \frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{4}. Nā, tāpiria te pūrua o te \frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia \frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=-\frac{5}{2}+\frac{49}{16}$$

Tāpiri -\frac{5}{2} ki te \frac{49}{16} 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.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{9}{16}$$

Tauwehea x^{2}+\frac{7}{2}x+\frac{49}{16}. 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}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{9}{16}$$

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

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Whakarūnātia.

$$x+\frac{7}{4}=\frac{3}{4} x+\frac{7}{4}=-\frac{3}{4}$$

Me tango \frac{7}{4} mai i ngā taha e rua o te whārite.

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