Whakaoti i te -2x^2+6x+16=-4

$$- 2 x ^ { 2 } + 6 x + 16 = - 4$$

Whakaoti mō x

$$x=-2$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 4 ki ngā taha e rua.

$$-2x^{2}+6x+16+4=0$$

Tāpirihia te 16 ki te 4, ka 20.

$$-2x^{2}+6x+20=0$$

Whakawehea ngā taha e rua ki te 2.

$$-x^{2}+3x+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=-10=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -10.

$$-1,10 -2,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+10=9 -2+5=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=5 b=-2$$

Tuhia anō te -x^{2}+3x+10 hei \left(-x^{2}+5x\right)+\left(-2x+10\right).

$$\left(-x^{2}+5x\right)+\left(-2x+10\right)$$

Tauwehea te -x i te tuatahi me te -2 i te rōpū tuarua.

$$-x\left(x-5\right)-2\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(-x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te -x-2=0.

$$x=5 x=-2$$

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.

$$-2x^{2}+6x+16=-4$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-2x^{2}+6x+16-\left(-4\right)=-4-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+6x+16-\left(-4\right)=0$$

Tango -4 mai i 16.

$$-2x^{2}+6x+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 6 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-6±\sqrt{36+8\times 20}}{2\left(-2\right)}$$

Whakareatia 8 ki te 20.

$$x=\frac{-6±\sqrt{36+160}}{2\left(-2\right)}$$

Tāpiri 36 ki te 160.

$$x=\frac{-6±\sqrt{196}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-6±14}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-6±14}{-4}$$

Nā, me whakaoti te whārite x=\frac{-6±14}{-4} ina he tāpiri te ±. Tāpiri -6 ki te 14.

$$x=\frac{8}{-4}$$

Whakawehe 8 ki te -4.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-6±14}{-4} ina he tango te ±. Tango 14 mai i -6.

$$x=-\frac{20}{-4}$$

Whakawehe -20 ki te -4.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=-2 x=5$$

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.

$$-2x^{2}+6x+16=-4$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-2x^{2}+6x+16-16=-4-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+6x=-4-16$$

Tango 16 mai i -4.

$$-2x^{2}+6x=-20$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+6x}{-2}=-\frac{20}{-2}$$

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

$$x^{2}+\frac{6}{-2}x=-\frac{20}{-2}$$

Whakawehe 6 ki te -2.

$$x^{2}-3x=-\frac{20}{-2}$$

Whakawehe -20 ki te -2.

$$x^{2}-3x=10$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=10+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{49}{4}$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{7}{2} x-\frac{3}{2}=-\frac{7}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=5 x=-2$$