Whakaoti i te -4(x)=2x^2+7x-4

$$-4(x)=2 { x }^{ 2 } +7x-4$$

Whakaoti mō x

$$x=\frac{3\sqrt{17}-11}{4}\approx 0.342329219$$

$$x=\frac{-3\sqrt{17}-11}{4}\approx -5.842329219$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 2x^{2} mai i ngā taha e rua.

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

Tangohia te 7x mai i ngā taha e rua.

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

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

$$-11x-2x^{2}=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$-11x-2x^{2}+4=0$$

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.

$$-2x^{2}-11x+4=0$$

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

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

Pūrua -11.

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

Whakareatia -4 ki te -2.

$$x=\frac{-\left(-11\right)±\sqrt{121+8\times 4}}{2\left(-2\right)}$$

Whakareatia 8 ki te 4.

$$x=\frac{-\left(-11\right)±\sqrt{121+32}}{2\left(-2\right)}$$

Tāpiri 121 ki te 32.

$$x=\frac{-\left(-11\right)±\sqrt{153}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 153.

$$x=\frac{-\left(-11\right)±3\sqrt{17}}{2\left(-2\right)}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±3\sqrt{17}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{11±3\sqrt{17}}{-4}$$

Nā, me whakaoti te whārite x=\frac{11±3\sqrt{17}}{-4} ina he tāpiri te ±. Tāpiri 11 ki te 3\sqrt{17}.

$$x=\frac{3\sqrt{17}+11}{-4}$$

Whakawehe 11+3\sqrt{17} ki te -4.

$$x=\frac{-3\sqrt{17}-11}{4}$$

Nā, me whakaoti te whārite x=\frac{11±3\sqrt{17}}{-4} ina he tango te ±. Tango 3\sqrt{17} mai i 11.

$$x=\frac{11-3\sqrt{17}}{-4}$$

Whakawehe 11-3\sqrt{17} ki te -4.

$$x=\frac{3\sqrt{17}-11}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3\sqrt{17}-11}{4} x=\frac{3\sqrt{17}-11}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 2x^{2} mai i ngā taha e rua.

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

Tangohia te 7x mai i ngā taha e rua.

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

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

$$-11x-2x^{2}=-4$$

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.

$$-2x^{2}-11x=-4$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}-11x}{-2}=-\frac{4}{-2}$$

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

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

Whakawehe -11 ki te -2.

$$x^{2}+\frac{11}{2}x=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$x^{2}+\frac{11}{2}x=2$$

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.

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

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

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

Tāpiri 2 ki te \frac{121}{16}.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{153}{16}$$

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

$$\left(x+\frac{11}{4}\right)^{2}=\frac{153}{16}$$

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

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{153}{16}}$$

Whakarūnātia.

$$x+\frac{11}{4}=\frac{3\sqrt{17}}{4} x+\frac{11}{4}=-\frac{3\sqrt{17}}{4}$$

Me tango \frac{11}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{17}-11}{4} x=\frac{-3\sqrt{17}-11}{4}$$