Whakaoti i te x^2+40x-75=0

$$x ^ { 2 } + 40 x - 75 = 0$$

Whakaoti mō x

$$x=5\sqrt{19}-20\approx 1.794494718$$

$$x=-5\sqrt{19}-20\approx -41.794494718$$

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.

$$x^{2}+40x-75=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 40 mō b, me -75 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-75\right)}}{2}$$

Pūrua 40.

$$x=\frac{-40±\sqrt{1600-4\left(-75\right)}}{2}$$

Whakareatia -4 ki te -75.

$$x=\frac{-40±\sqrt{1600+300}}{2}$$

Tāpiri 1600 ki te 300.

$$x=\frac{-40±\sqrt{1900}}{2}$$

Tuhia te pūtakerua o te 1900.

$$x=\frac{-40±10\sqrt{19}}{2}$$

Nā, me whakaoti te whārite x=\frac{-40±10\sqrt{19}}{2} ina he tāpiri te ±. Tāpiri -40 ki te 10\sqrt{19}.

$$x=\frac{10\sqrt{19}-40}{2}$$

Whakawehe -40+10\sqrt{19} ki te 2.

$$x=5\sqrt{19}-20$$

Nā, me whakaoti te whārite x=\frac{-40±10\sqrt{19}}{2} ina he tango te ±. Tango 10\sqrt{19} mai i -40.

$$x=\frac{-10\sqrt{19}-40}{2}$$

Whakawehe -40-10\sqrt{19} ki te 2.

$$x=-5\sqrt{19}-20$$

Kua oti te whārite te whakatau.

$$x=5\sqrt{19}-20 x=-5\sqrt{19}-20$$

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.

$$x^{2}+40x-75=0$$

Me tāpiri 75 ki ngā taha e rua o te whārite.

$$x^{2}+40x-75-\left(-75\right)=-\left(-75\right)$$

Mā te tango i te -75 i a ia ake anō ka toe ko te 0.

$$x^{2}+40x=-\left(-75\right)$$

Tango -75 mai i 0.

$$x^{2}+40x=75$$

Whakawehea te 40, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 20. Nā, tāpiria te pūrua o te 20 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+40x+20^{2}=75+20^{2}$$

Pūrua 20.

$$x^{2}+40x+400=75+400$$

Tāpiri 75 ki te 400.

$$x^{2}+40x+400=475$$

Tauwehea x^{2}+40x+400. 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}.

$$\left(x+20\right)^{2}=475$$

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

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{475}$$

Whakarūnātia.

$$x+20=5\sqrt{19} x+20=-5\sqrt{19}$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$x=5\sqrt{19}-20 x=-5\sqrt{19}-20$$