Whakaoti i te x^2-115x=550

$$x ^ { 2 } - 115 x = 550$$

Whakaoti mō x

$$x = \frac{5 \sqrt{617} + 115}{2} \approx 119.598711742$$

$$x=\frac{115-5\sqrt{617}}{2}\approx -4.598711742$$

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.

$$x^{2}-115x=550$$

Me tango 550 mai i ngā taha e rua o te whārite.

$$x^{2}-115x-550=550-550$$

Mā te tango i te 550 i a ia ake anō ka toe ko te 0.

$$x^{2}-115x-550=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -115 mō b, me -550 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-115\right)±\sqrt{\left(-115\right)^{2}-4\left(-550\right)}}{2}$$

Pūrua -115.

$$x=\frac{-\left(-115\right)±\sqrt{13225-4\left(-550\right)}}{2}$$

Whakareatia -4 ki te -550.

$$x=\frac{-\left(-115\right)±\sqrt{13225+2200}}{2}$$

Tāpiri 13225 ki te 2200.

$$x=\frac{-\left(-115\right)±\sqrt{15425}}{2}$$

Tuhia te pūtakerua o te 15425.

$$x=\frac{-\left(-115\right)±5\sqrt{617}}{2}$$

Ko te tauaro o -115 ko 115.

$$x=\frac{115±5\sqrt{617}}{2}$$

Nā, me whakaoti te whārite x=\frac{115±5\sqrt{617}}{2} ina he tāpiri te ±. Tāpiri 115 ki te 5\sqrt{617}.

$$x=\frac{5\sqrt{617}+115}{2}$$

Nā, me whakaoti te whārite x=\frac{115±5\sqrt{617}}{2} ina he tango te ±. Tango 5\sqrt{617} mai i 115.

$$x=\frac{115-5\sqrt{617}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{617}+115}{2} x=\frac{115-5\sqrt{617}}{2}$$

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.

$$x^{2}-115x=550$$

Whakawehea te -115, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{115}{2}. Nā, tāpiria te pūrua o te -\frac{115}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-115x+\left(-\frac{115}{2}\right)^{2}=550+\left(-\frac{115}{2}\right)^{2}$$

Pūruatia -\frac{115}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-115x+\frac{13225}{4}=550+\frac{13225}{4}$$

Tāpiri 550 ki te \frac{13225}{4}.

$$x^{2}-115x+\frac{13225}{4}=\frac{15425}{4}$$

Tauwehea te x^{2}-115x+\frac{13225}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{115}{2}\right)^{2}=\frac{15425}{4}$$

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

$$\sqrt{\left(x-\frac{115}{2}\right)^{2}}=\sqrt{\frac{15425}{4}}$$

Whakarūnātia.

$$x-\frac{115}{2}=\frac{5\sqrt{617}}{2} x-\frac{115}{2}=-\frac{5\sqrt{617}}{2}$$

Me tāpiri \frac{115}{2} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{617}+115}{2} x=\frac{115-5\sqrt{617}}{2}$$