Whakaoti i te -x^2+2x+15=0

$$- x ^ { 2 } + 2 x + 15 = 0$$

Whakaoti mō x

$$x=-3$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=-15=-15$$

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 -15.

$$-1,15 -3,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+15=14 -3+5=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=5 b=-3$$

Tuhia anō te -x^{2}+2x+15 hei \left(-x^{2}+5x\right)+\left(-3x+15\right).

$$\left(-x^{2}+5x\right)+\left(-3x+15\right)$$

Tauwehea te -x i te tuatahi me te -3 i te rōpū tuarua.

$$-x\left(x-5\right)-3\left(x-5\right)$$

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

$$\left(x-5\right)\left(-x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te -x-3=0.

$$x=5 x=-3$$

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.

$$-x^{2}+2x+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 2 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-2±\sqrt{4+4\times 15}}{2\left(-1\right)}$$

Whakareatia 4 ki te 15.

$$x=\frac{-2±\sqrt{4+60}}{2\left(-1\right)}$$

Tāpiri 4 ki te 60.

$$x=\frac{-2±\sqrt{64}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-2±8}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-2±8}{-2}$$

Nā, me whakaoti te whārite x=\frac{-2±8}{-2} ina he tāpiri te ±. Tāpiri -2 ki te 8.

$$x=\frac{6}{-2}$$

Whakawehe 6 ki te -2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-2±8}{-2} ina he tango te ±. Tango 8 mai i -2.

$$x=-\frac{10}{-2}$$

Whakawehe -10 ki te -2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=-3 x=5$$

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.

$$-x^{2}+2x+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-x^{2}+2x+15-15=-15$$

Mā te tango i te 15 i a ia ake anō ka toe ko te 0.

$$-x^{2}+2x=-15$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{15}{-1}$$

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

$$x^{2}+\frac{2}{-1}x=-\frac{15}{-1}$$

Whakawehe 2 ki te -1.

$$x^{2}-2x=-\frac{15}{-1}$$

Whakawehe -15 ki te -1.

$$x^{2}-2x=15$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=15+1$$

Tāpiri 15 ki te 1.

$$x^{2}-2x+1=16$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=16$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$x-1=4 x-1=-4$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=5 x=-3$$