Whakaoti i te x^2+20x=45

$$x ^ { 2 } + 20 x = 45$$

Whakaoti mō x (complex solution)

$$x=\sqrt{145}-10\approx 2.041594579$$

$$x=-\left(\sqrt{145}+10\right)\approx -22.041594579$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}+20x=45$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$x^{2}+20x-45=45-45$$

Mā te tango i te 45 i a ia ake anō ka toe ko te 0.

$$x^{2}+20x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 20 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-45\right)}}{2}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-45\right)}}{2}$$

Whakareatia -4 ki te -45.

$$x=\frac{-20±\sqrt{400+180}}{2}$$

Tāpiri 400 ki te 180.

$$x=\frac{-20±\sqrt{580}}{2}$$

Tuhia te pūtakerua o te 580.

$$x=\frac{-20±2\sqrt{145}}{2}$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{145}}{2} ina he tāpiri te ±. Tāpiri -20 ki te 2\sqrt{145}.

$$x=\frac{2\sqrt{145}-20}{2}$$

Whakawehe -20+2\sqrt{145} ki te 2.

$$x=\sqrt{145}-10$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{145}}{2} ina he tango te ±. Tango 2\sqrt{145} mai i -20.

$$x=\frac{-2\sqrt{145}-20}{2}$$

Whakawehe -20-2\sqrt{145} ki te 2.

$$x=-\sqrt{145}-10$$

Kua oti te whārite te whakatau.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}+20x=45$$

Whakawehea te 20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 10. Nā, tāpiria te pūrua o te 10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+20x+10^{2}=45+10^{2}$$

Pūrua 10.

$$x^{2}+20x+100=45+100$$

Tāpiri 45 ki te 100.

$$x^{2}+20x+100=145$$

Tauwehea x^{2}+20x+100. 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}.

$$\left(x+10\right)^{2}=145$$

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

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{145}$$

Whakarūnātia.

$$x+10=\sqrt{145} x+10=-\sqrt{145}$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

Whakaoti mō x

$$x=\sqrt{145}-10\approx 2.041594579$$

$$x=-\sqrt{145}-10\approx -22.041594579$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}+20x=45$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$x^{2}+20x-45=45-45$$

Mā te tango i te 45 i a ia ake anō ka toe ko te 0.

$$x^{2}+20x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 20 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-45\right)}}{2}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-45\right)}}{2}$$

Whakareatia -4 ki te -45.

$$x=\frac{-20±\sqrt{400+180}}{2}$$

Tāpiri 400 ki te 180.

$$x=\frac{-20±\sqrt{580}}{2}$$

Tuhia te pūtakerua o te 580.

$$x=\frac{-20±2\sqrt{145}}{2}$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{145}}{2} ina he tāpiri te ±. Tāpiri -20 ki te 2\sqrt{145}.

$$x=\frac{2\sqrt{145}-20}{2}$$

Whakawehe -20+2\sqrt{145} ki te 2.

$$x=\sqrt{145}-10$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{145}}{2} ina he tango te ±. Tango 2\sqrt{145} mai i -20.

$$x=\frac{-2\sqrt{145}-20}{2}$$

Whakawehe -20-2\sqrt{145} ki te 2.

$$x=-\sqrt{145}-10$$

Kua oti te whārite te whakatau.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}+20x=45$$

Whakawehea te 20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 10. Nā, tāpiria te pūrua o te 10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+20x+10^{2}=45+10^{2}$$

Pūrua 10.

$$x^{2}+20x+100=45+100$$

Tāpiri 45 ki te 100.

$$x^{2}+20x+100=145$$

Tauwehea x^{2}+20x+100. 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}.

$$\left(x+10\right)^{2}=145$$

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

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{145}$$

Whakarūnātia.

$$x+10=\sqrt{145} x+10=-\sqrt{145}$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$