Whakaoti i te 2x^2+17x+21=0

$$2 x ^ { 2 } + 17 x + 21 = 0$$

Whakaoti mō x

$$x=-7$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=2\times 21=42$$

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 42.

$$1,42 2,21 3,14 6,7$$

Tātaihia te tapeke mō ia takirua.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=3 b=14$$

Tuhia anō te 2x^{2}+17x+21 hei \left(2x^{2}+3x\right)+\left(14x+21\right).

$$\left(2x^{2}+3x\right)+\left(14x+21\right)$$

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

$$x\left(2x+3\right)+7\left(2x+3\right)$$

Whakatauwehea atu te kīanga pātahi 2x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x+3\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te 2x+3=0 me te x+7=0.

$$x=-\frac{3}{2} x=-7$$

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.

$$2x^{2}+17x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 17 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 2\times 21}}{2\times 2}$$

Pūrua 17.

$$x=\frac{-17±\sqrt{289-4\times 2\times 21}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-17±\sqrt{289-8\times 21}}{2\times 2}$$

Whakareatia -8 ki te 21.

$$x=\frac{-17±\sqrt{289-168}}{2\times 2}$$

Tāpiri 289 ki te -168.

$$x=\frac{-17±\sqrt{121}}{2\times 2}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-17±11}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-17±11}{4}$$

Nā, me whakaoti te whārite x=\frac{-17±11}{4} ina he tāpiri te ±. Tāpiri -17 ki te 11.

$$x=-\frac{6}{4}$$

Whakahekea te hautanga \frac{-6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{-17±11}{4} ina he tango te ±. Tango 11 mai i -17.

$$x=-\frac{28}{4}$$

Whakawehe -28 ki te 4.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=-\frac{3}{2} x=-7$$

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.

$$2x^{2}+17x+21=0$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$2x^{2}+17x+21-21=-21$$

Mā te tango i te 21 i a ia ake anō ka toe ko te 0.

$$2x^{2}+17x=-21$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+17x}{2}=-\frac{21}{2}$$

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

$$x^{2}+\frac{17}{2}x=-\frac{21}{2}$$

Whakawehea te \frac{17}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{17}{4}. Nā, tāpiria te pūrua o te \frac{17}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{17}{2}x+\left(\frac{17}{4}\right)^{2}=-\frac{21}{2}+\left(\frac{17}{4}\right)^{2}$$

Pūruatia \frac{17}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{17}{2}x+\frac{289}{16}=-\frac{21}{2}+\frac{289}{16}$$

Tāpiri -\frac{21}{2} ki te \frac{289}{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.

$$x^{2}+\frac{17}{2}x+\frac{289}{16}=\frac{121}{16}$$

Tauwehea x^{2}+\frac{17}{2}x+\frac{289}{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}.

$$\left(x+\frac{17}{4}\right)^{2}=\frac{121}{16}$$

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

$$\sqrt{\left(x+\frac{17}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Whakarūnātia.

$$x+\frac{17}{4}=\frac{11}{4} x+\frac{17}{4}=-\frac{11}{4}$$

Me tango \frac{17}{4} mai i ngā taha e rua o te whārite.

$$x=-\frac{3}{2} x=-7$$