Whakaoti i te 35*15=(19-x)(15+x)

$$35 \times 15=(19-x)(15+x)$$

Whakaoti mō x (complex solution)

$$x=2+2\sqrt{59}i\approx 2+15.362291496i$$

$$x=-2\sqrt{59}i+2\approx 2-15.362291496i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 35 ki te 15, ka 525.

$$525=\left(19-x\right)\left(15+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 19-x ki te 15+x ka whakakotahi i ngā kupu rite.

$$525=285+4x-x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$285+4x-x^{2}=525$$

Tangohia te 525 mai i ngā taha e rua.

$$285+4x-x^{2}-525=0$$

Tangohia te 525 i te 285, ka -240.

$$-240+4x-x^{2}=0$$

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.

$$-x^{2}+4x-240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 4 mō b, me -240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-4±\sqrt{16+4\left(-240\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -240.

$$x=\frac{-4±\sqrt{16-960}}{2\left(-1\right)}$$

Tāpiri 16 ki te -960.

$$x=\frac{-4±\sqrt{-944}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -944.

$$x=\frac{-4±4\sqrt{59}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-4±4\sqrt{59}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-4±4\sqrt{59}i}{-2} ina he tāpiri te ±. Tāpiri -4 ki te 4i\sqrt{59}.

$$x=\frac{-4+4\sqrt{59}i}{-2}$$

Whakawehe -4+4i\sqrt{59} ki te -2.

$$x=-2\sqrt{59}i+2$$

Nā, me whakaoti te whārite x=\frac{-4±4\sqrt{59}i}{-2} ina he tango te ±. Tango 4i\sqrt{59} mai i -4.

$$x=\frac{-4\sqrt{59}i-4}{-2}$$

Whakawehe -4-4i\sqrt{59} ki te -2.

$$x=2+2\sqrt{59}i$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{59}i+2 x=2+2\sqrt{59}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 35 ki te 15, ka 525.

$$525=\left(19-x\right)\left(15+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 19-x ki te 15+x ka whakakotahi i ngā kupu rite.

$$525=285+4x-x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$285+4x-x^{2}=525$$

Tangohia te 285 mai i ngā taha e rua.

$$4x-x^{2}=525-285$$

Tangohia te 285 i te 525, ka 240.

$$4x-x^{2}=240$$

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.

$$-x^{2}+4x=240$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+4x}{-1}=\frac{240}{-1}$$

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

$$x^{2}+\frac{4}{-1}x=\frac{240}{-1}$$

Whakawehe 4 ki te -1.

$$x^{2}-4x=\frac{240}{-1}$$

Whakawehe 240 ki te -1.

$$x^{2}-4x=-240$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-240+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=-240+4$$

Tāpiri -240 ki te 4.

$$x^{2}-4x+4=-236$$

Tauwehea x^{2}-4x+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}.

$$\left(x-2\right)^{2}=-236$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-236}$$

Whakarūnātia.

$$x-2=2\sqrt{59}i x-2=-2\sqrt{59}i$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=2+2\sqrt{59}i x=-2\sqrt{59}i+2$$