Whakaoti i te 15.30quadx^2-30x-470=0

$$15.30 \quad x ^ { 2 } - 30 x - 470 = 0$$

Whakaoti mō x

$$x = \frac{20 \sqrt{206} + 50}{51} \approx 6.608901998$$

$$x=\frac{50-20\sqrt{206}}{51}\approx -4.648117684$$

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.

$$15.3x^{2}-30x-470=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15.3 mō a, -30 mō b, me -470 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 15.3\left(-470\right)}}{2\times 15.3}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 15.3\left(-470\right)}}{2\times 15.3}$$

Whakareatia -4 ki te 15.3.

$$x=\frac{-\left(-30\right)±\sqrt{900-61.2\left(-470\right)}}{2\times 15.3}$$

Whakareatia -61.2 ki te -470.

$$x=\frac{-\left(-30\right)±\sqrt{900+28764}}{2\times 15.3}$$

Tāpiri 900 ki te 28764.

$$x=\frac{-\left(-30\right)±\sqrt{29664}}{2\times 15.3}$$

Tuhia te pūtakerua o te 29664.

$$x=\frac{-\left(-30\right)±12\sqrt{206}}{2\times 15.3}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±12\sqrt{206}}{2\times 15.3}$$

Whakareatia 2 ki te 15.3.

$$x=\frac{30±12\sqrt{206}}{30.6}$$

Nā, me whakaoti te whārite x=\frac{30±12\sqrt{206}}{30.6} ina he tāpiri te ±. Tāpiri 30 ki te 12\sqrt{206}.

$$x=\frac{12\sqrt{206}+30}{30.6}$$

Whakawehe 30+12\sqrt{206} ki te 30.6 mā te whakarea 30+12\sqrt{206} ki te tau huripoki o 30.6.

$$x=\frac{20\sqrt{206}+50}{51}$$

Nā, me whakaoti te whārite x=\frac{30±12\sqrt{206}}{30.6} ina he tango te ±. Tango 12\sqrt{206} mai i 30.

$$x=\frac{30-12\sqrt{206}}{30.6}$$

Whakawehe 30-12\sqrt{206} ki te 30.6 mā te whakarea 30-12\sqrt{206} ki te tau huripoki o 30.6.

$$x=\frac{50-20\sqrt{206}}{51}$$

Kua oti te whārite te whakatau.

$$x=\frac{20\sqrt{206}+50}{51} x=\frac{50-20\sqrt{206}}{51}$$

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.

$$15.3x^{2}-30x-470=0$$

Me tāpiri 470 ki ngā taha e rua o te whārite.

$$15.3x^{2}-30x-470-\left(-470\right)=-\left(-470\right)$$

Mā te tango i te -470 i a ia ake anō ka toe ko te 0.

$$15.3x^{2}-30x=-\left(-470\right)$$

Tango -470 mai i 0.

$$15.3x^{2}-30x=470$$

Whakawehea ngā taha e rua o te whārite ki te 15.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{15.3x^{2}-30x}{15.3}=\frac{470}{15.3}$$

Mā te whakawehe ki te 15.3 ka wetekia te whakareanga ki te 15.3.

$$x^{2}+\left(-\frac{30}{15.3}\right)x=\frac{470}{15.3}$$

Whakawehe -30 ki te 15.3 mā te whakarea -30 ki te tau huripoki o 15.3.

$$x^{2}-\frac{100}{51}x=\frac{470}{15.3}$$

Whakawehe 470 ki te 15.3 mā te whakarea 470 ki te tau huripoki o 15.3.

$$x^{2}-\frac{100}{51}x=\frac{4700}{153}$$

Whakawehea te -\frac{100}{51}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{50}{51}. Nā, tāpiria te pūrua o te -\frac{50}{51} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{100}{51}x+\left(-\frac{50}{51}\right)^{2}=\frac{4700}{153}+\left(-\frac{50}{51}\right)^{2}$$

Pūruatia -\frac{50}{51} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{100}{51}x+\frac{2500}{2601}=\frac{4700}{153}+\frac{2500}{2601}$$

Tāpiri \frac{4700}{153} ki te \frac{2500}{2601} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{100}{51}x+\frac{2500}{2601}=\frac{82400}{2601}$$

Tauwehea x^{2}-\frac{100}{51}x+\frac{2500}{2601}. 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}.

$$\left(x-\frac{50}{51}\right)^{2}=\frac{82400}{2601}$$

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

$$\sqrt{\left(x-\frac{50}{51}\right)^{2}}=\sqrt{\frac{82400}{2601}}$$

Whakarūnātia.

$$x-\frac{50}{51}=\frac{20\sqrt{206}}{51} x-\frac{50}{51}=-\frac{20\sqrt{206}}{51}$$

Me tāpiri \frac{50}{51} ki ngā taha e rua o te whārite.

$$x=\frac{20\sqrt{206}+50}{51} x=\frac{50-20\sqrt{206}}{51}$$