Whakaoti i te 22x^2+24x-9=0

$$22 x ^ { 2 } + 24 x - 9 = 0$$

Whakaoti mō x

$$x=\frac{3\sqrt{38}}{22}-\frac{6}{11}\approx 0.295147364$$

$$x=-\frac{3\sqrt{38}}{22}-\frac{6}{11}\approx -1.386056455$$

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.

$$22x^{2}+24x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 22 mō a, 24 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 22\left(-9\right)}}{2\times 22}$$

Pūrua 24.

$$x=\frac{-24±\sqrt{576-4\times 22\left(-9\right)}}{2\times 22}$$

Whakareatia -4 ki te 22.

$$x=\frac{-24±\sqrt{576-88\left(-9\right)}}{2\times 22}$$

Whakareatia -88 ki te -9.

$$x=\frac{-24±\sqrt{576+792}}{2\times 22}$$

Tāpiri 576 ki te 792.

$$x=\frac{-24±\sqrt{1368}}{2\times 22}$$

Tuhia te pūtakerua o te 1368.

$$x=\frac{-24±6\sqrt{38}}{2\times 22}$$

Whakareatia 2 ki te 22.

$$x=\frac{-24±6\sqrt{38}}{44}$$

Nā, me whakaoti te whārite x=\frac{-24±6\sqrt{38}}{44} ina he tāpiri te ±. Tāpiri -24 ki te 6\sqrt{38}.

$$x=\frac{6\sqrt{38}-24}{44}$$

Whakawehe -24+6\sqrt{38} ki te 44.

$$x=\frac{3\sqrt{38}}{22}-\frac{6}{11}$$

Nā, me whakaoti te whārite x=\frac{-24±6\sqrt{38}}{44} ina he tango te ±. Tango 6\sqrt{38} mai i -24.

$$x=\frac{-6\sqrt{38}-24}{44}$$

Whakawehe -24-6\sqrt{38} ki te 44.

$$x=-\frac{3\sqrt{38}}{22}-\frac{6}{11}$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{38}}{22}-\frac{6}{11} x=-\frac{3\sqrt{38}}{22}-\frac{6}{11}$$

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.

$$22x^{2}+24x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$22x^{2}+24x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$22x^{2}+24x=-\left(-9\right)$$

Tango -9 mai i 0.

$$22x^{2}+24x=9$$

Whakawehea ngā taha e rua ki te 22.

$$\frac{22x^{2}+24x}{22}=\frac{9}{22}$$

Mā te whakawehe ki te 22 ka wetekia te whakareanga ki te 22.

$$x^{2}+\frac{24}{22}x=\frac{9}{22}$$

Whakahekea te hautanga \frac{24}{22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{12}{11}x=\frac{9}{22}$$

Whakawehea te \frac{12}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{6}{11}. Nā, tāpiria te pūrua o te \frac{6}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{12}{11}x+\left(\frac{6}{11}\right)^{2}=\frac{9}{22}+\left(\frac{6}{11}\right)^{2}$$

Pūruatia \frac{6}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{12}{11}x+\frac{36}{121}=\frac{9}{22}+\frac{36}{121}$$

Tāpiri \frac{9}{22} ki te \frac{36}{121} 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.

$$x^{2}+\frac{12}{11}x+\frac{36}{121}=\frac{171}{242}$$

Tauwehea x^{2}+\frac{12}{11}x+\frac{36}{121}. 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}.

$$\left(x+\frac{6}{11}\right)^{2}=\frac{171}{242}$$

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

$$\sqrt{\left(x+\frac{6}{11}\right)^{2}}=\sqrt{\frac{171}{242}}$$

Whakarūnātia.

$$x+\frac{6}{11}=\frac{3\sqrt{38}}{22} x+\frac{6}{11}=-\frac{3\sqrt{38}}{22}$$

Me tango \frac{6}{11} mai i ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{38}}{22}-\frac{6}{11} x=-\frac{3\sqrt{38}}{22}-\frac{6}{11}$$