Whakaoti i te x^2+4x-45=0

$$x ^ { 2 } + 4 x - 45 = 0$$

Whakaoti mō x

$$x=-9$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+4x-45 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.

$$a+b=4 ab=-45$$

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 -45.

$$-1,45 -3,15 -5,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+45=44 -3+15=12 -5+9=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-5 b=9$$

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

$$\left(x-5\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+9=0.

$$x=5 x=-9$$

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-45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=1\left(-45\right)=-45$$

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 -45.

$$-1,45 -3,15 -5,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+45=44 -3+15=12 -5+9=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-5 b=9$$

Tuhia anō te x^{2}+4x-45 hei \left(x^{2}-5x\right)+\left(9x-45\right).

$$\left(x^{2}-5x\right)+\left(9x-45\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(x-5\right)+9\left(x-5\right)$$

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

$$\left(x-5\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+9=0.

$$x=5 x=-9$$

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.

$$x^{2}+4x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 4 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-45\right)}}{2}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-45\right)}}{2}$$

Whakareatia -4 ki te -45.

$$x=\frac{-4±\sqrt{16+180}}{2}$$

Tāpiri 16 ki te 180.

$$x=\frac{-4±\sqrt{196}}{2}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-4±14}{2}$$

Nā, me whakaoti te whārite x=\frac{-4±14}{2} ina he tāpiri te ±. Tāpiri -4 ki te 14.

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

Whakawehe 10 ki te 2.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{-4±14}{2} ina he tango te ±. Tango 14 mai i -4.

$$x=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$x=-9$$

Kua oti te whārite te whakatau.

$$x=5 x=-9$$

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.

$$x^{2}+4x-45=0$$

Me tāpiri 45 ki ngā taha e rua o te whārite.

$$x^{2}+4x-45-\left(-45\right)=-\left(-45\right)$$

Mā te tango i te -45 i a ia ake anō ka toe ko te 0.

$$x^{2}+4x=-\left(-45\right)$$

Tango -45 mai i 0.

$$x^{2}+4x=45$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=45+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=45+4$$

Tāpiri 45 ki te 4.

$$x^{2}+4x+4=49$$

Tauwehea te x^{2}+4x+4. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

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

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

Whakarūnātia.

$$x+2=7 x+2=-7$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=5 x=-9$$