Whakaoti i te 2x^2-7x-2=4x+5

$$2 x ^ { 2 } - 7 x - 2 = 4 x + 5$$

Whakaoti mō x

$$x = \frac{\sqrt{177} + 11}{4} \approx 6.076033674$$

$$x=\frac{11-\sqrt{177}}{4}\approx -0.576033674$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 4x mai i ngā taha e rua.

$$2x^{2}-7x-2-4x=5$$

Pahekotia te -7x me -4x, ka -11x.

$$2x^{2}-11x-2=5$$

Tangohia te 5 mai i ngā taha e rua.

$$2x^{2}-11x-2-5=0$$

Tangohia te 5 i te -2, ka -7.

$$2x^{2}-11x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -11 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 2\left(-7\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-11\right)±\sqrt{121-8\left(-7\right)}}{2\times 2}$$

Whakareatia -8 ki te -7.

$$x=\frac{-\left(-11\right)±\sqrt{121+56}}{2\times 2}$$

Tāpiri 121 ki te 56.

$$x=\frac{-\left(-11\right)±\sqrt{177}}{2\times 2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±\sqrt{177}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{11±\sqrt{177}}{4}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{177}}{4} ina he tāpiri te ±. Tāpiri 11 ki te \sqrt{177}.

$$x=\frac{\sqrt{177}+11}{4}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{177}}{4} ina he tango te ±. Tango \sqrt{177} mai i 11.

$$x=\frac{11-\sqrt{177}}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{177}+11}{4} x=\frac{11-\sqrt{177}}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 4x mai i ngā taha e rua.

$$2x^{2}-7x-2-4x=5$$

Pahekotia te -7x me -4x, ka -11x.

$$2x^{2}-11x-2=5$$

Me tāpiri te 2 ki ngā taha e rua.

$$2x^{2}-11x=5+2$$

Tāpirihia te 5 ki te 2, ka 7.

$$2x^{2}-11x=7$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-11x}{2}=\frac{7}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}-\frac{11}{2}x=\frac{7}{2}$$

Whakawehea te -\frac{11}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{4}. Nā, tāpiria te pūrua o te -\frac{11}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=\frac{7}{2}+\left(-\frac{11}{4}\right)^{2}$$

Pūruatia -\frac{11}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{7}{2}+\frac{121}{16}$$

Tāpiri \frac{7}{2} ki te \frac{121}{16} 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.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{177}{16}$$

Tauwehea te x^{2}-\frac{11}{2}x+\frac{121}{16}. 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}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{177}{16}$$

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

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{177}{16}}$$

Whakarūnātia.

$$x-\frac{11}{4}=\frac{\sqrt{177}}{4} x-\frac{11}{4}=-\frac{\sqrt{177}}{4}$$

Me tāpiri \frac{11}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{177}+11}{4} x=\frac{11-\sqrt{177}}{4}$$