Whakaoti i te x^2-9x-70=0

$${ x }^{ 2 } -9x-70 = 0$$

Whakaoti mō x

$$x=-5$$

$$x=14$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-9x-70 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.

$$a+b=-9 ab=-70$$

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 -70.

$$1,-70 2,-35 5,-14 7,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-70=-69 2-35=-33 5-14=-9 7-10=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-14 b=5$$

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

$$\left(x-14\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-14=0 me te x+5=0.

$$x=14 x=-5$$

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-70. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=1\left(-70\right)=-70$$

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 -70.

$$1,-70 2,-35 5,-14 7,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-70=-69 2-35=-33 5-14=-9 7-10=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-14 b=5$$

Tuhia anō te x^{2}-9x-70 hei \left(x^{2}-14x\right)+\left(5x-70\right).

$$\left(x^{2}-14x\right)+\left(5x-70\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x-14\right)+5\left(x-14\right)$$

Whakatauwehea atu te kīanga pātahi x-14 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-14\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-14=0 me te x+5=0.

$$x=14 x=-5$$

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.

$$x^{2}-9x-70=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -9 mō b, me -70 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-70\right)}}{2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-70\right)}}{2}$$

Whakareatia -4 ki te -70.

$$x=\frac{-\left(-9\right)±\sqrt{81+280}}{2}$$

Tāpiri 81 ki te 280.

$$x=\frac{-\left(-9\right)±\sqrt{361}}{2}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-\left(-9\right)±19}{2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±19}{2}$$

Nā, me whakaoti te whārite x=\frac{9±19}{2} ina he tāpiri te ±. Tāpiri 9 ki te 19.

$$x=\frac{28}{2}$$

Whakawehe 28 ki te 2.

$$x=14$$

Nā, me whakaoti te whārite x=\frac{9±19}{2} ina he tango te ±. Tango 19 mai i 9.

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

Whakawehe -10 ki te 2.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=14 x=-5$$

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.

$$x^{2}-9x-70=0$$

Me tāpiri 70 ki ngā taha e rua o te whārite.

$$x^{2}-9x-70-\left(-70\right)=-\left(-70\right)$$

Mā te tango i te -70 i a ia ake anō ka toe ko te 0.

$$x^{2}-9x=-\left(-70\right)$$

Tango -70 mai i 0.

$$x^{2}-9x=70$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=70+\left(-\frac{9}{2}\right)^{2}$$

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

$$x^{2}-9x+\frac{81}{4}=70+\frac{81}{4}$$

Tāpiri 70 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{361}{4}$$

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

$$\left(x-\frac{9}{2}\right)^{2}=\frac{361}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{19}{2} x-\frac{9}{2}=-\frac{19}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=14 x=-5$$