Whakaoti i te (6-x*x)+(7*x)=30

$$(6-x \times x)+(7 \times x)=30$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{47}i+7}{2}\approx 3.5-3.4278273i$$

$$x=\frac{7+\sqrt{47}i}{2}\approx 3.5+3.4278273i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te x ki te x, ka x^{2}.

$$6-x^{2}+7x=30$$

Tangohia te 30 mai i ngā taha e rua.

$$6-x^{2}+7x-30=0$$

Tangohia te 30 i te 6, ka -24.

$$-24-x^{2}+7x=0$$

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.

$$-x^{2}+7x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 7 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-7±\sqrt{49+4\left(-24\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -24.

$$x=\frac{-7±\sqrt{49-96}}{2\left(-1\right)}$$

Tāpiri 49 ki te -96.

$$x=\frac{-7±\sqrt{-47}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -47.

$$x=\frac{-7±\sqrt{47}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-7±\sqrt{47}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{47}i}{-2} ina he tāpiri te ±. Tāpiri -7 ki te i\sqrt{47}.

$$x=\frac{-7+\sqrt{47}i}{-2}$$

Whakawehe -7+i\sqrt{47} ki te -2.

$$x=\frac{-\sqrt{47}i+7}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{47}i}{-2} ina he tango te ±. Tango i\sqrt{47} mai i -7.

$$x=\frac{-\sqrt{47}i-7}{-2}$$

Whakawehe -7-i\sqrt{47} ki te -2.

$$x=\frac{7+\sqrt{47}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{47}i+7}{2} x=\frac{7+\sqrt{47}i}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te x ki te x, ka x^{2}.

$$6-x^{2}+7x=30$$

Tangohia te 6 mai i ngā taha e rua.

$$-x^{2}+7x=30-6$$

Tangohia te 6 i te 30, ka 24.

$$-x^{2}+7x=24$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+7x}{-1}=\frac{24}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{7}{-1}x=\frac{24}{-1}$$

Whakawehe 7 ki te -1.

$$x^{2}-7x=\frac{24}{-1}$$

Whakawehe 24 ki te -1.

$$x^{2}-7x=-24$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-24+\left(-\frac{7}{2}\right)^{2}$$

Pūruatia -\frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-7x+\frac{49}{4}=-24+\frac{49}{4}$$

Tāpiri -24 ki te \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=-\frac{47}{4}$$

Tauwehea x^{2}-7x+\frac{49}{4}. 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}.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{47}{4}$$

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

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{47}{4}}$$

Whakarūnātia.

$$x-\frac{7}{2}=\frac{\sqrt{47}i}{2} x-\frac{7}{2}=-\frac{\sqrt{47}i}{2}$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$x=\frac{7+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i+7}{2}$$