Whakaoti i te x^2-18x+7x-126=0

$${ x }^{ 2 } -18x+7x-126=0$$

Whakaoti mō x

$$x=-7$$

$$x=18$$

Ngā Upane Whakamahi Tauwehe

Pahekotia te -18x me 7x, ka -11x.

$$x^{2}-11x-126=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-11x-126 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.

$$a+b=-11 ab=-126$$

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 -126.

$$1,-126 2,-63 3,-42 6,-21 7,-18 9,-14$$

Tātaihia te tapeke mō ia takirua.

$$1-126=-125 2-63=-61 3-42=-39 6-21=-15 7-18=-11 9-14=-5$$

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

$$a=-18 b=7$$

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

$$\left(x-18\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te x+7=0.

$$x=18 x=-7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Pahekotia te -18x me 7x, ka -11x.

$$x^{2}-11x-126=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-126. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=1\left(-126\right)=-126$$

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 -126.

$$1,-126 2,-63 3,-42 6,-21 7,-18 9,-14$$

Tātaihia te tapeke mō ia takirua.

$$1-126=-125 2-63=-61 3-42=-39 6-21=-15 7-18=-11 9-14=-5$$

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

$$a=-18 b=7$$

Tuhia anō te x^{2}-11x-126 hei \left(x^{2}-18x\right)+\left(7x-126\right).

$$\left(x^{2}-18x\right)+\left(7x-126\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(x-18\right)+7\left(x-18\right)$$

Whakatauwehea atu te kīanga pātahi x-18 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-18\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te x+7=0.

$$x=18 x=-7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te -18x me 7x, ka -11x.

$$x^{2}-11x-126=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -11 mō b, me -126 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-126\right)}}{2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-126\right)}}{2}$$

Whakareatia -4 ki te -126.

$$x=\frac{-\left(-11\right)±\sqrt{121+504}}{2}$$

Tāpiri 121 ki te 504.

$$x=\frac{-\left(-11\right)±\sqrt{625}}{2}$$

Tuhia te pūtakerua o te 625.

$$x=\frac{-\left(-11\right)±25}{2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±25}{2}$$

Nā, me whakaoti te whārite x=\frac{11±25}{2} ina he tāpiri te ±. Tāpiri 11 ki te 25.

$$x=\frac{36}{2}$$

Whakawehe 36 ki te 2.

$$x=18$$

Nā, me whakaoti te whārite x=\frac{11±25}{2} ina he tango te ±. Tango 25 mai i 11.

$$x=-\frac{14}{2}$$

Whakawehe -14 ki te 2.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=18 x=-7$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te -18x me 7x, ka -11x.

$$x^{2}-11x-126=0$$

Me tāpiri te 126 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}-11x=126$$

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.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=126+\left(-\frac{11}{2}\right)^{2}$$

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

$$x^{2}-11x+\frac{121}{4}=126+\frac{121}{4}$$

Tāpiri 126 ki te \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{625}{4}$$

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

$$\left(x-\frac{11}{2}\right)^{2}=\frac{625}{4}$$

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

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Whakarūnātia.

$$x-\frac{11}{2}=\frac{25}{2} x-\frac{11}{2}=-\frac{25}{2}$$

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

$$x=18 x=-7$$