Whakaoti i te -x^2-5x=-1/2x+2

$$- x ^ { 2 } - 5 x = - \frac { 1 } { 2 } x + 2$$

Whakaoti mō x

$$x=-\frac{1}{2}=-0.5$$

$$x=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te \frac{1}{2}x ki ngā taha e rua.

$$-x^{2}-5x+\frac{1}{2}x=2$$

Pahekotia te -5x me \frac{1}{2}x, ka -\frac{9}{2}x.

$$-x^{2}-\frac{9}{2}x=2$$

Tangohia te 2 mai i ngā taha e rua.

$$-x^{2}-\frac{9}{2}x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -\frac{9}{2} mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\left(-\frac{9}{2}\right)^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}+4\left(-2\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -2.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-8}}{2\left(-1\right)}$$

Tāpiri \frac{81}{4} ki te -8.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{49}{4}}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te \frac{49}{4}.

$$x=\frac{-\left(-\frac{9}{2}\right)±\frac{7}{2}}{2\left(-1\right)}$$

Ko te tauaro o -\frac{9}{2} ko \frac{9}{2}.

$$x=\frac{\frac{9}{2}±\frac{7}{2}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{\frac{9}{2}±\frac{7}{2}}{-2}$$

Nā, me whakaoti te whārite x=\frac{\frac{9}{2}±\frac{7}{2}}{-2} ina he tāpiri te ±. Tāpiri \frac{9}{2} ki te \frac{7}{2} 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.

$$x=\frac{8}{-2}$$

Whakawehe 8 ki te -2.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{\frac{9}{2}±\frac{7}{2}}{-2} ina he tango te ±. Tango \frac{7}{2} mai i \frac{9}{2} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1}{-2}$$

Whakawehe 1 ki te -2.

$$x=-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=-4 x=-\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te \frac{1}{2}x ki ngā taha e rua.

$$-x^{2}-5x+\frac{1}{2}x=2$$

Pahekotia te -5x me \frac{1}{2}x, ka -\frac{9}{2}x.

$$-x^{2}-\frac{9}{2}x=2$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-\frac{9}{2}x}{-1}=\frac{2}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{\frac{9}{2}}{-1}\right)x=\frac{2}{-1}$$

Whakawehe -\frac{9}{2} ki te -1.

$$x^{2}+\frac{9}{2}x=\frac{2}{-1}$$

Whakawehe 2 ki te -1.

$$x^{2}+\frac{9}{2}x=-2$$

Whakawehea te \frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{4}. Nā, tāpiria te pūrua o te \frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=-2+\left(\frac{9}{4}\right)^{2}$$

Pūruatia \frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=-2+\frac{81}{16}$$

Tāpiri -2 ki te \frac{81}{16}.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{49}{16}$$

Tauwehea x^{2}+\frac{9}{2}x+\frac{81}{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}.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{49}{16}$$

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

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Whakarūnātia.

$$x+\frac{9}{4}=\frac{7}{4} x+\frac{9}{4}=-\frac{7}{4}$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{2} x=-4$$