Whakaoti i te z^2+3z-30=(2z+5)(z+6)

$$z ^ { 2 } + 3 z - 30 = ( 2 z + 5 ) ( z + 6 )$$

Whakaoti mō z

$$z=-\sqrt{11}i-7\approx -7-3.31662479i$$

$$z=-7+\sqrt{11}i\approx -7+3.31662479i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2z+5 ki te z+6 ka whakakotahi i ngā kupu rite.

$$z^{2}+3z-30=2z^{2}+17z+30$$

Tangohia te 2z^{2} mai i ngā taha e rua.

$$z^{2}+3z-30-2z^{2}=17z+30$$

Pahekotia te z^{2} me -2z^{2}, ka -z^{2}.

$$-z^{2}+3z-30=17z+30$$

Tangohia te 17z mai i ngā taha e rua.

$$-z^{2}+3z-30-17z=30$$

Pahekotia te 3z me -17z, ka -14z.

$$-z^{2}-14z-30=30$$

Tangohia te 30 mai i ngā taha e rua.

$$-z^{2}-14z-30-30=0$$

Tangohia te 30 i te -30, ka -60.

$$-z^{2}-14z-60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -14 mō b, me -60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

Pūrua -14.

$$z=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$z=\frac{-\left(-14\right)±\sqrt{196+4\left(-60\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -60.

$$z=\frac{-\left(-14\right)±\sqrt{196-240}}{2\left(-1\right)}$$

Tāpiri 196 ki te -240.

$$z=\frac{-\left(-14\right)±\sqrt{-44}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -44.

$$z=\frac{-\left(-14\right)±2\sqrt{11}i}{2\left(-1\right)}$$

Ko te tauaro o -14 ko 14.

$$z=\frac{14±2\sqrt{11}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$z=\frac{14±2\sqrt{11}i}{-2}$$

Nā, me whakaoti te whārite z=\frac{14±2\sqrt{11}i}{-2} ina he tāpiri te ±. Tāpiri 14 ki te 2i\sqrt{11}.

$$z=\frac{14+2\sqrt{11}i}{-2}$$

Whakawehe 14+2i\sqrt{11} ki te -2.

$$z=-\sqrt{11}i-7$$

Nā, me whakaoti te whārite z=\frac{14±2\sqrt{11}i}{-2} ina he tango te ±. Tango 2i\sqrt{11} mai i 14.

$$z=\frac{-2\sqrt{11}i+14}{-2}$$

Whakawehe 14-2i\sqrt{11} ki te -2.

$$z=-7+\sqrt{11}i$$

Kua oti te whārite te whakatau.

$$z=-\sqrt{11}i-7 z=-7+\sqrt{11}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2z+5 ki te z+6 ka whakakotahi i ngā kupu rite.

$$z^{2}+3z-30=2z^{2}+17z+30$$

Tangohia te 2z^{2} mai i ngā taha e rua.

$$z^{2}+3z-30-2z^{2}=17z+30$$

Pahekotia te z^{2} me -2z^{2}, ka -z^{2}.

$$-z^{2}+3z-30=17z+30$$

Tangohia te 17z mai i ngā taha e rua.

$$-z^{2}+3z-30-17z=30$$

Pahekotia te 3z me -17z, ka -14z.

$$-z^{2}-14z-30=30$$

Me tāpiri te 30 ki ngā taha e rua.

$$-z^{2}-14z=30+30$$

Tāpirihia te 30 ki te 30, ka 60.

$$-z^{2}-14z=60$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-z^{2}-14z}{-1}=\frac{60}{-1}$$

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

$$z^{2}+\left(-\frac{14}{-1}\right)z=\frac{60}{-1}$$

Whakawehe -14 ki te -1.

$$z^{2}+14z=\frac{60}{-1}$$

Whakawehe 60 ki te -1.

$$z^{2}+14z=-60$$

Whakawehea te 14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 7. Nā, tāpiria te pūrua o te 7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}+14z+7^{2}=-60+7^{2}$$

Pūrua 7.

$$z^{2}+14z+49=-60+49$$

Tāpiri -60 ki te 49.

$$z^{2}+14z+49=-11$$

Tauwehea z^{2}+14z+49. 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}.

$$\left(z+7\right)^{2}=-11$$

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

$$\sqrt{\left(z+7\right)^{2}}=\sqrt{-11}$$

Whakarūnātia.

$$z+7=\sqrt{11}i z+7=-\sqrt{11}i$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$z=-7+\sqrt{11}i z=-\sqrt{11}i-7$$