Whakaoti i te x^2-6x-40=0

$$x ^ { 2 } - 6 x - 40 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x-40 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=-40$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-10 b=4$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

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

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

$$x=10 x=-4$$

Ngā Upane 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.

$$a+b=-6 ab=1\left(-40\right)=-40$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-10 b=4$$

Tuhia anō te x^{2}-6x-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.

$$x^{2}-6x-40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me -40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-40\right)}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-40\right)}}{2}$$

Whakareatia -4 ki te -40.

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

Tāpiri 36 ki te 160.

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

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-6\right)±14}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±14}{2}$$

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

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

Whakawehe 20 ki te 2.

$$x=10$$

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

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

Whakawehe -8 ki te 2.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=10 x=-4$$

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.

$$x^{2}-6x-40=0$$

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

$$x^{2}-6x-40-\left(-40\right)=-\left(-40\right)$$

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

$$x^{2}-6x=-\left(-40\right)$$

Tango -40 mai i 0.

$$x^{2}-6x=40$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=40+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=40+9$$

Tāpiri 40 ki te 9.

$$x^{2}-6x+9=49$$

Tauwehea x^{2}-6x+9. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=10 x=-4$$