Whakaoti i te x^2-22x-23=0

$${ x }^{ 2 } -22x-23=0$$

Whakaoti mō x

$$x=-1$$

$$x=23$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-22x-23 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.

$$a+b=-22 ab=-23$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-23 b=1$$

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

$$\left(x-23\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-23=0 me te x+1=0.

$$x=23 x=-1$$

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-23. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-22 ab=1\left(-23\right)=-23$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-23 b=1$$

Tuhia anō te x^{2}-22x-23 hei \left(x^{2}-23x\right)+\left(x-23\right).

$$\left(x^{2}-23x\right)+\left(x-23\right)$$

Whakatauwehea atu x i te x^{2}-23x.

$$x\left(x-23\right)+x-23$$

Whakatauwehea atu te kīanga pātahi x-23 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-23\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-23=0 me te x+1=0.

$$x=23 x=-1$$

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.

$$x^{2}-22x-23=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -22 mō b, me -23 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-23\right)}}{2}$$

Pūrua -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\left(-23\right)}}{2}$$

Whakareatia -4 ki te -23.

$$x=\frac{-\left(-22\right)±\sqrt{484+92}}{2}$$

Tāpiri 484 ki te 92.

$$x=\frac{-\left(-22\right)±\sqrt{576}}{2}$$

Tuhia te pūtakerua o te 576.

$$x=\frac{-\left(-22\right)±24}{2}$$

Ko te tauaro o -22 ko 22.

$$x=\frac{22±24}{2}$$

Nā, me whakaoti te whārite x=\frac{22±24}{2} ina he tāpiri te ±. Tāpiri 22 ki te 24.

$$x=\frac{46}{2}$$

Whakawehe 46 ki te 2.

$$x=23$$

Nā, me whakaoti te whārite x=\frac{22±24}{2} ina he tango te ±. Tango 24 mai i 22.

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

Whakawehe -2 ki te 2.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=23 x=-1$$

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.

$$x^{2}-22x-23=0$$

Me tāpiri 23 ki ngā taha e rua o te whārite.

$$x^{2}-22x-23-\left(-23\right)=-\left(-23\right)$$

Mā te tango i te -23 i a ia ake anō ka toe ko te 0.

$$x^{2}-22x=-\left(-23\right)$$

Tango -23 mai i 0.

$$x^{2}-22x=23$$

Whakawehea te -22, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -11. Nā, tāpiria te pūrua o te -11 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-22x+\left(-11\right)^{2}=23+\left(-11\right)^{2}$$

Pūrua -11.

$$x^{2}-22x+121=23+121$$

Tāpiri 23 ki te 121.

$$x^{2}-22x+121=144$$

Tauwehea x^{2}-22x+121. 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}.

$$\left(x-11\right)^{2}=144$$

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

$$\sqrt{\left(x-11\right)^{2}}=\sqrt{144}$$

Whakarūnātia.

$$x-11=12 x-11=-12$$

Me tāpiri 11 ki ngā taha e rua o te whārite.

$$x=23 x=-1$$