Whakaoti i te ((20-a)(380+5a)=32000

$$( ( 20 - a ) ( 380 + 5 a ) = 32000$$

Whakaoti mō a

$$a=-28+64i$$

$$a=-28-64i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-a ki te 380+5a ka whakakotahi i ngā kupu rite.

$$7600-280a-5a^{2}=32000$$

Tangohia te 32000 mai i ngā taha e rua.

$$7600-280a-5a^{2}-32000=0$$

Tangohia te 32000 i te 7600, ka -24400.

$$-24400-280a-5a^{2}=0$$

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.

$$-5a^{2}-280a-24400=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, -280 mō b, me -24400 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-280\right)±\sqrt{\left(-280\right)^{2}-4\left(-5\right)\left(-24400\right)}}{2\left(-5\right)}$$

Pūrua -280.

$$a=\frac{-\left(-280\right)±\sqrt{78400-4\left(-5\right)\left(-24400\right)}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$a=\frac{-\left(-280\right)±\sqrt{78400+20\left(-24400\right)}}{2\left(-5\right)}$$

Whakareatia 20 ki te -24400.

$$a=\frac{-\left(-280\right)±\sqrt{78400-488000}}{2\left(-5\right)}$$

Tāpiri 78400 ki te -488000.

$$a=\frac{-\left(-280\right)±\sqrt{-409600}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te -409600.

$$a=\frac{-\left(-280\right)±640i}{2\left(-5\right)}$$

Ko te tauaro o -280 ko 280.

$$a=\frac{280±640i}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$a=\frac{280±640i}{-10}$$

Nā, me whakaoti te whārite a=\frac{280±640i}{-10} ina he tāpiri te ±. Tāpiri 280 ki te 640i.

$$a=\frac{280+640i}{-10}$$

Whakawehe 280+640i ki te -10.

$$a=-28-64i$$

Nā, me whakaoti te whārite a=\frac{280±640i}{-10} ina he tango te ±. Tango 640i mai i 280.

$$a=\frac{280-640i}{-10}$$

Whakawehe 280-640i ki te -10.

$$a=-28+64i$$

Kua oti te whārite te whakatau.

$$a=-28-64i a=-28+64i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-a ki te 380+5a ka whakakotahi i ngā kupu rite.

$$7600-280a-5a^{2}=32000$$

Tangohia te 7600 mai i ngā taha e rua.

$$-280a-5a^{2}=32000-7600$$

Tangohia te 7600 i te 32000, ka 24400.

$$-280a-5a^{2}=24400$$

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.

$$-5a^{2}-280a=24400$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5a^{2}-280a}{-5}=\frac{24400}{-5}$$

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

$$a^{2}+\left(-\frac{280}{-5}\right)a=\frac{24400}{-5}$$

Whakawehe -280 ki te -5.

$$a^{2}+56a=\frac{24400}{-5}$$

Whakawehe 24400 ki te -5.

$$a^{2}+56a=-4880$$

Whakawehea te 56, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 28. Nā, tāpiria te pūrua o te 28 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+56a+28^{2}=-4880+28^{2}$$

Pūrua 28.

$$a^{2}+56a+784=-4880+784$$

Tāpiri -4880 ki te 784.

$$a^{2}+56a+784=-4096$$

Tauwehea a^{2}+56a+784. 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}.

$$\left(a+28\right)^{2}=-4096$$

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

$$\sqrt{\left(a+28\right)^{2}}=\sqrt{-4096}$$

Whakarūnātia.

$$a+28=64i a+28=-64i$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$a=-28+64i a=-28-64i$$