Whakaoti i te x^2=11x+12

$$x ^ { 2 } = 11 x + 12$$

Whakaoti mō x

$$x=-1$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 11x mai i ngā taha e rua.

$$x^{2}-11x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$x^{2}-11x-12=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-11x-12 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.

$$a+b=-11 ab=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-12 b=1$$

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

$$\left(x-12\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+1=0.

$$x=12 x=-1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 11x mai i ngā taha e rua.

$$x^{2}-11x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$x^{2}-11x-12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=1\left(-12\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-12 b=1$$

Tuhia anō te x^{2}-11x-12 hei \left(x^{2}-12x\right)+\left(x-12\right).

$$\left(x^{2}-12x\right)+\left(x-12\right)$$

Whakatauwehea atu x i te x^{2}-12x.

$$x\left(x-12\right)+x-12$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+1=0.

$$x=12 x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 11x mai i ngā taha e rua.

$$x^{2}-11x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$x^{2}-11x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -11 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-12\right)}}{2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-12\right)}}{2}$$

Whakareatia -4 ki te -12.

$$x=\frac{-\left(-11\right)±\sqrt{121+48}}{2}$$

Tāpiri 121 ki te 48.

$$x=\frac{-\left(-11\right)±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-11\right)±13}{2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±13}{2}$$

Nā, me whakaoti te whārite x=\frac{11±13}{2} ina he tāpiri te ±. Tāpiri 11 ki te 13.

$$x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{11±13}{2} ina he tango te ±. Tango 13 mai i 11.

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

Whakawehe -2 ki te 2.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=12 x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 11x mai i ngā taha e rua.

$$x^{2}-11x=12$$

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

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=12+\left(-\frac{11}{2}\right)^{2}$$

Pūruatia -\frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-11x+\frac{121}{4}=12+\frac{121}{4}$$

Tāpiri 12 ki te \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{169}{4}$$

Tauwehea te x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x-\frac{11}{2}=\frac{13}{2} x-\frac{11}{2}=-\frac{13}{2}$$

Me tāpiri \frac{11}{2} ki ngā taha e rua o te whārite.

$$x=12 x=-1$$