Whakaoti i te 7x^2+2=30x-10

$$7 { x }^{ 2 } +2=30x-10$$

Whakaoti mō x

$$x = \frac{\sqrt{141} + 15}{7} \approx 3.839191727$$

$$x=\frac{15-\sqrt{141}}{7}\approx 0.446522559$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 30x mai i ngā taha e rua.

$$7x^{2}+2-30x=-10$$

Me tāpiri te 10 ki ngā taha e rua.

$$7x^{2}+2-30x+10=0$$

Tāpirihia te 2 ki te 10, ka 12.

$$7x^{2}+12-30x=0$$

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.

$$7x^{2}-30x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -30 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 7\times 12}}{2\times 7}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 7\times 12}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-30\right)±\sqrt{900-28\times 12}}{2\times 7}$$

Whakareatia -28 ki te 12.

$$x=\frac{-\left(-30\right)±\sqrt{900-336}}{2\times 7}$$

Tāpiri 900 ki te -336.

$$x=\frac{-\left(-30\right)±\sqrt{564}}{2\times 7}$$

Tuhia te pūtakerua o te 564.

$$x=\frac{-\left(-30\right)±2\sqrt{141}}{2\times 7}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±2\sqrt{141}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{30±2\sqrt{141}}{14}$$

Nā, me whakaoti te whārite x=\frac{30±2\sqrt{141}}{14} ina he tāpiri te ±. Tāpiri 30 ki te 2\sqrt{141}.

$$x=\frac{2\sqrt{141}+30}{14}$$

Whakawehe 30+2\sqrt{141} ki te 14.

$$x=\frac{\sqrt{141}+15}{7}$$

Nā, me whakaoti te whārite x=\frac{30±2\sqrt{141}}{14} ina he tango te ±. Tango 2\sqrt{141} mai i 30.

$$x=\frac{30-2\sqrt{141}}{14}$$

Whakawehe 30-2\sqrt{141} ki te 14.

$$x=\frac{15-\sqrt{141}}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{141}+15}{7} x=\frac{15-\sqrt{141}}{7}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 30x mai i ngā taha e rua.

$$7x^{2}+2-30x=-10$$

Tangohia te 2 mai i ngā taha e rua.

$$7x^{2}-30x=-10-2$$

Tangohia te 2 i te -10, ka -12.

$$7x^{2}-30x=-12$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-30x}{7}=-\frac{12}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}-\frac{30}{7}x=-\frac{12}{7}$$

Whakawehea te -\frac{30}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{7}. Nā, tāpiria te pūrua o te -\frac{15}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{30}{7}x+\left(-\frac{15}{7}\right)^{2}=-\frac{12}{7}+\left(-\frac{15}{7}\right)^{2}$$

Pūruatia -\frac{15}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{30}{7}x+\frac{225}{49}=-\frac{12}{7}+\frac{225}{49}$$

Tāpiri -\frac{12}{7} ki te \frac{225}{49} 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.

$$x^{2}-\frac{30}{7}x+\frac{225}{49}=\frac{141}{49}$$

Tauwehea te x^{2}-\frac{30}{7}x+\frac{225}{49}. 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}.

$$\left(x-\frac{15}{7}\right)^{2}=\frac{141}{49}$$

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

$$\sqrt{\left(x-\frac{15}{7}\right)^{2}}=\sqrt{\frac{141}{49}}$$

Whakarūnātia.

$$x-\frac{15}{7}=\frac{\sqrt{141}}{7} x-\frac{15}{7}=-\frac{\sqrt{141}}{7}$$

Me tāpiri \frac{15}{7} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{141}+15}{7} x=\frac{15-\sqrt{141}}{7}$$