Whakaoti i te -x^2+14x-40=0

$$- x ^ { 2 } + 14 x - 40 = 0$$

Whakaoti mō x

$$x=4$$

$$x=10$$

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-40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=-\left(-40\right)=40$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 40.

$$1,40 2,20 4,10 5,8$$

Tātaihia te tapeke mō ia takirua.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=10 b=4$$

Tuhia anō te -x^{2}+14x-40 hei \left(-x^{2}+10x\right)+\left(4x-40\right).

$$\left(-x^{2}+10x\right)+\left(4x-40\right)$$

Tauwehea te -x i te tuatahi me te 4 i te rōpū tuarua.

$$-x\left(x-10\right)+4\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(-x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te -x+4=0.

$$x=10 x=4$$

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.

$$-x^{2}+14x-40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 14 mō b, me -40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-40\right)}}{2\left(-1\right)}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-40\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-14±\sqrt{196+4\left(-40\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -40.

$$x=\frac{-14±\sqrt{196-160}}{2\left(-1\right)}$$

Tāpiri 196 ki te -160.

$$x=\frac{-14±\sqrt{36}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-14±6}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-14±6}{-2}$$

Nā, me whakaoti te whārite x=\frac{-14±6}{-2} ina he tāpiri te ±. Tāpiri -14 ki te 6.

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

Whakawehe -8 ki te -2.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-14±6}{-2} ina he tango te ±. Tango 6 mai i -14.

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

Whakawehe -20 ki te -2.

$$x=10$$

Kua oti te whārite te whakatau.

$$x=4 x=10$$

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.

$$-x^{2}+14x-40=0$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$-x^{2}+14x-40-\left(-40\right)=-\left(-40\right)$$

Mā te tango i te -40 i a ia ake anō ka toe ko te 0.

$$-x^{2}+14x=-\left(-40\right)$$

Tango -40 mai i 0.

$$-x^{2}+14x=40$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+14x}{-1}=\frac{40}{-1}$$

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

$$x^{2}+\frac{14}{-1}x=\frac{40}{-1}$$

Whakawehe 14 ki te -1.

$$x^{2}-14x=\frac{40}{-1}$$

Whakawehe 40 ki te -1.

$$x^{2}-14x=-40$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-40+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-40+49$$

Tāpiri -40 ki te 49.

$$x^{2}-14x+49=9$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=9$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-7=3 x-7=-3$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=10 x=4$$