Whakaoti i te 4078379=4040a-a^2

$$4078379 = 4040 a - a ^ { 2 }$$

Whakaoti mō a

$$a=\sqrt{2021}+2020\approx 2064.955533586$$

$$a=2020-\sqrt{2021}\approx 1975.044466414$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$4040a-a^{2}=4078379$$

Tangohia te 4078379 mai i ngā taha e rua.

$$4040a-a^{2}-4078379=0$$

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.

$$-a^{2}+4040a-4078379=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 4040 mō b, me -4078379 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-4040±\sqrt{4040^{2}-4\left(-1\right)\left(-4078379\right)}}{2\left(-1\right)}$$

Pūrua 4040.

$$a=\frac{-4040±\sqrt{16321600-4\left(-1\right)\left(-4078379\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$a=\frac{-4040±\sqrt{16321600+4\left(-4078379\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -4078379.

$$a=\frac{-4040±\sqrt{16321600-16313516}}{2\left(-1\right)}$$

Tāpiri 16321600 ki te -16313516.

$$a=\frac{-4040±\sqrt{8084}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 8084.

$$a=\frac{-4040±2\sqrt{2021}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$a=\frac{-4040±2\sqrt{2021}}{-2}$$

Nā, me whakaoti te whārite a=\frac{-4040±2\sqrt{2021}}{-2} ina he tāpiri te ±. Tāpiri -4040 ki te 2\sqrt{2021}.

$$a=\frac{2\sqrt{2021}-4040}{-2}$$

Whakawehe -4040+2\sqrt{2021} ki te -2.

$$a=2020-\sqrt{2021}$$

Nā, me whakaoti te whārite a=\frac{-4040±2\sqrt{2021}}{-2} ina he tango te ±. Tango 2\sqrt{2021} mai i -4040.

$$a=\frac{-2\sqrt{2021}-4040}{-2}$$

Whakawehe -4040-2\sqrt{2021} ki te -2.

$$a=\sqrt{2021}+2020$$

Kua oti te whārite te whakatau.

$$a=2020-\sqrt{2021} a=\sqrt{2021}+2020$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$4040a-a^{2}=4078379$$

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.

$$-a^{2}+4040a=4078379$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-a^{2}+4040a}{-1}=\frac{4078379}{-1}$$

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

$$a^{2}+\frac{4040}{-1}a=\frac{4078379}{-1}$$

Whakawehe 4040 ki te -1.

$$a^{2}-4040a=\frac{4078379}{-1}$$

Whakawehe 4078379 ki te -1.

$$a^{2}-4040a=-4078379$$

Whakawehea te -4040, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2020. Nā, tāpiria te pūrua o te -2020 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-4040a+\left(-2020\right)^{2}=-4078379+\left(-2020\right)^{2}$$

Pūrua -2020.

$$a^{2}-4040a+4080400=-4078379+4080400$$

Tāpiri -4078379 ki te 4080400.

$$a^{2}-4040a+4080400=2021$$

Tauwehea a^{2}-4040a+4080400. 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}.

$$\left(a-2020\right)^{2}=2021$$

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

$$\sqrt{\left(a-2020\right)^{2}}=\sqrt{2021}$$

Whakarūnātia.

$$a-2020=\sqrt{2021} a-2020=-\sqrt{2021}$$

Me tāpiri 2020 ki ngā taha e rua o te whārite.

$$a=\sqrt{2021}+2020 a=2020-\sqrt{2021}$$