Whakaoti i te 4x^2=7-9x

$$4 x ^ { 2 } = 7 - 9 x$$

Whakaoti mō x

$$x=\frac{\sqrt{193}-9}{8}\approx 0.611555499$$

$$x=\frac{-\sqrt{193}-9}{8}\approx -2.861555499$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 7 mai i ngā taha e rua.

$$4x^{2}-7=-9x$$

Me tāpiri te 9x ki ngā taha e rua.

$$4x^{2}-7+9x=0$$

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.

$$4x^{2}+9x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 9 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 4\left(-7\right)}}{2\times 4}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 4\left(-7\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-9±\sqrt{81-16\left(-7\right)}}{2\times 4}$$

Whakareatia -16 ki te -7.

$$x=\frac{-9±\sqrt{81+112}}{2\times 4}$$

Tāpiri 81 ki te 112.

$$x=\frac{-9±\sqrt{193}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-9±\sqrt{193}}{8}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{193}}{8} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{193}.

$$x=\frac{\sqrt{193}-9}{8}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{193}}{8} ina he tango te ±. Tango \sqrt{193} mai i -9.

$$x=\frac{-\sqrt{193}-9}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{193}-9}{8} x=\frac{-\sqrt{193}-9}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 9x ki ngā taha e rua.

$$4x^{2}+9x=7$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+9x}{4}=\frac{7}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\frac{9}{4}x=\frac{7}{4}$$

Whakawehea te \frac{9}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{8}. Nā, tāpiria te pūrua o te \frac{9}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{9}{4}x+\left(\frac{9}{8}\right)^{2}=\frac{7}{4}+\left(\frac{9}{8}\right)^{2}$$

Pūruatia \frac{9}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{7}{4}+\frac{81}{64}$$

Tāpiri \frac{7}{4} ki te \frac{81}{64} 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.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{193}{64}$$

Tauwehea x^{2}+\frac{9}{4}x+\frac{81}{64}. 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}.

$$\left(x+\frac{9}{8}\right)^{2}=\frac{193}{64}$$

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

$$\sqrt{\left(x+\frac{9}{8}\right)^{2}}=\sqrt{\frac{193}{64}}$$

Whakarūnātia.

$$x+\frac{9}{8}=\frac{\sqrt{193}}{8} x+\frac{9}{8}=-\frac{\sqrt{193}}{8}$$

Me tango \frac{9}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{193}-9}{8} x=\frac{-\sqrt{193}-9}{8}$$