Whakaoti i te (x)=3(x-5)(x+3)

$$( x ) = 3 ( x - 5 ) ( x + 3 )$$

Whakaoti mō x

$$x = \frac{\sqrt{589} + 7}{6} \approx 5.2115537$$

$$x=\frac{7-\sqrt{589}}{6}\approx -2.878220367$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-5.

$$x=\left(3x-15\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-15 ki te x+3 ka whakakotahi i ngā kupu rite.

$$x=3x^{2}-6x-45$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$x-3x^{2}=-6x-45$$

Me tāpiri te 6x ki ngā taha e rua.

$$x-3x^{2}+6x=-45$$

Pahekotia te x me 6x, ka 7x.

$$7x-3x^{2}=-45$$

Me tāpiri te 45 ki ngā taha e rua.

$$7x-3x^{2}+45=0$$

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.

$$-3x^{2}+7x+45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 7 mō b, me 45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\times 45}}{2\left(-3\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\times 45}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-7±\sqrt{49+12\times 45}}{2\left(-3\right)}$$

Whakareatia 12 ki te 45.

$$x=\frac{-7±\sqrt{49+540}}{2\left(-3\right)}$$

Tāpiri 49 ki te 540.

$$x=\frac{-7±\sqrt{589}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-7±\sqrt{589}}{-6}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{589}}{-6} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{589}.

$$x=\frac{\sqrt{589}-7}{-6}$$

Whakawehe -7+\sqrt{589} ki te -6.

$$x=\frac{7-\sqrt{589}}{6}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{589}}{-6} ina he tango te ±. Tango \sqrt{589} mai i -7.

$$x=\frac{-\sqrt{589}-7}{-6}$$

Whakawehe -7-\sqrt{589} ki te -6.

$$x=\frac{\sqrt{589}+7}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{7-\sqrt{589}}{6} x=\frac{\sqrt{589}+7}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-5.

$$x=\left(3x-15\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-15 ki te x+3 ka whakakotahi i ngā kupu rite.

$$x=3x^{2}-6x-45$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$x-3x^{2}=-6x-45$$

Me tāpiri te 6x ki ngā taha e rua.

$$x-3x^{2}+6x=-45$$

Pahekotia te x me 6x, ka 7x.

$$7x-3x^{2}=-45$$

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.

$$-3x^{2}+7x=-45$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+7x}{-3}=-\frac{45}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{7}{-3}x=-\frac{45}{-3}$$

Whakawehe 7 ki te -3.

$$x^{2}-\frac{7}{3}x=-\frac{45}{-3}$$

Whakawehe -45 ki te -3.

$$x^{2}-\frac{7}{3}x=15$$

Whakawehea te -\frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{6}. Nā, tāpiria te pūrua o te -\frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=15+\left(-\frac{7}{6}\right)^{2}$$

Pūruatia -\frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=15+\frac{49}{36}$$

Tāpiri 15 ki te \frac{49}{36}.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{589}{36}$$

Tauwehea x^{2}-\frac{7}{3}x+\frac{49}{36}. 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}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{589}{36}$$

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

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{589}{36}}$$

Whakarūnātia.

$$x-\frac{7}{6}=\frac{\sqrt{589}}{6} x-\frac{7}{6}=-\frac{\sqrt{589}}{6}$$

Me tāpiri \frac{7}{6} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{589}+7}{6} x=\frac{7-\sqrt{589}}{6}$$