Whakaoti i te 5x^2+10x-20=0

$$5 { x }^{ 2 } +10x-20=0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{5}-1\approx 1.236067977$$

$$x=-\left(\sqrt{5}+1\right)\approx -3.236067977$$

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.

$$5x^{2}+10x-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 10 mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-20\right)}}{2\times 5}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-20\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-10±\sqrt{100-20\left(-20\right)}}{2\times 5}$$

Whakareatia -20 ki te -20.

$$x=\frac{-10±\sqrt{100+400}}{2\times 5}$$

Tāpiri 100 ki te 400.

$$x=\frac{-10±\sqrt{500}}{2\times 5}$$

Tuhia te pūtakerua o te 500.

$$x=\frac{-10±10\sqrt{5}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-10±10\sqrt{5}}{10}$$

Nā, me whakaoti te whārite x=\frac{-10±10\sqrt{5}}{10} ina he tāpiri te ±. Tāpiri -10 ki te 10\sqrt{5}.

$$x=\frac{10\sqrt{5}-10}{10}$$

Whakawehe -10+10\sqrt{5} ki te 10.

$$x=\sqrt{5}-1$$

Nā, me whakaoti te whārite x=\frac{-10±10\sqrt{5}}{10} ina he tango te ±. Tango 10\sqrt{5} mai i -10.

$$x=\frac{-10\sqrt{5}-10}{10}$$

Whakawehe -10-10\sqrt{5} ki te 10.

$$x=-\sqrt{5}-1$$

Kua oti te whārite te whakatau.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$

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.

$$5x^{2}+10x-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$5x^{2}+10x-20-\left(-20\right)=-\left(-20\right)$$

Mā te tango i te -20 i a ia ake anō ka toe ko te 0.

$$5x^{2}+10x=-\left(-20\right)$$

Tango -20 mai i 0.

$$5x^{2}+10x=20$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+10x}{5}=\frac{20}{5}$$

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

$$x^{2}+\frac{10}{5}x=\frac{20}{5}$$

Whakawehe 10 ki te 5.

$$x^{2}+2x=\frac{20}{5}$$

Whakawehe 20 ki te 5.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=4+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=4+1$$

Tāpiri 4 ki te 1.

$$x^{2}+2x+1=5$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=5$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$x+1=\sqrt{5} x+1=-\sqrt{5}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$

Whakaoti mō x

$$x=\sqrt{5}-1\approx 1.236067977$$

$$x=-\sqrt{5}-1\approx -3.236067977$$

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.

$$5x^{2}+10x-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 10 mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-20\right)}}{2\times 5}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-20\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-10±\sqrt{100-20\left(-20\right)}}{2\times 5}$$

Whakareatia -20 ki te -20.

$$x=\frac{-10±\sqrt{100+400}}{2\times 5}$$

Tāpiri 100 ki te 400.

$$x=\frac{-10±\sqrt{500}}{2\times 5}$$

Tuhia te pūtakerua o te 500.

$$x=\frac{-10±10\sqrt{5}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-10±10\sqrt{5}}{10}$$

Nā, me whakaoti te whārite x=\frac{-10±10\sqrt{5}}{10} ina he tāpiri te ±. Tāpiri -10 ki te 10\sqrt{5}.

$$x=\frac{10\sqrt{5}-10}{10}$$

Whakawehe -10+10\sqrt{5} ki te 10.

$$x=\sqrt{5}-1$$

Nā, me whakaoti te whārite x=\frac{-10±10\sqrt{5}}{10} ina he tango te ±. Tango 10\sqrt{5} mai i -10.

$$x=\frac{-10\sqrt{5}-10}{10}$$

Whakawehe -10-10\sqrt{5} ki te 10.

$$x=-\sqrt{5}-1$$

Kua oti te whārite te whakatau.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$

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.

$$5x^{2}+10x-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$5x^{2}+10x-20-\left(-20\right)=-\left(-20\right)$$

Mā te tango i te -20 i a ia ake anō ka toe ko te 0.

$$5x^{2}+10x=-\left(-20\right)$$

Tango -20 mai i 0.

$$5x^{2}+10x=20$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+10x}{5}=\frac{20}{5}$$

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

$$x^{2}+\frac{10}{5}x=\frac{20}{5}$$

Whakawehe 10 ki te 5.

$$x^{2}+2x=\frac{20}{5}$$

Whakawehe 20 ki te 5.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=4+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=4+1$$

Tāpiri 4 ki te 1.

$$x^{2}+2x+1=5$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=5$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$x+1=\sqrt{5} x+1=-\sqrt{5}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$