Whakaoti i te x(x-4)+7(x-4)=0

$$x(x-4)+7(x-4)=0$$

Whakaoti mō x

$$x=-7$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-4.

$$x^{2}-4x+7\left(x-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$x^{2}-4x+7x-28=0$$

Pahekotia te -4x me 7x, ka 3x.

$$x^{2}+3x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+3x-28 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.

$$a+b=3 ab=-28$$

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 -28.

$$-1,28 -2,14 -4,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+28=27 -2+14=12 -4+7=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-4 b=7$$

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

$$\left(x-4\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+7=0.

$$x=4 x=-7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-4.

$$x^{2}-4x+7\left(x-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$x^{2}-4x+7x-28=0$$

Pahekotia te -4x me 7x, ka 3x.

$$x^{2}+3x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\left(-28\right)=-28$$

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 -28.

$$-1,28 -2,14 -4,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+28=27 -2+14=12 -4+7=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-4 b=7$$

Tuhia anō te x^{2}+3x-28 hei \left(x^{2}-4x\right)+\left(7x-28\right).

$$\left(x^{2}-4x\right)+\left(7x-28\right)$$

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

$$x\left(x-4\right)+7\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+7=0.

$$x=4 x=-7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-4.

$$x^{2}-4x+7\left(x-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$x^{2}-4x+7x-28=0$$

Pahekotia te -4x me 7x, ka 3x.

$$x^{2}+3x-28=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 3 mō b, me -28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-28\right)}}{2}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\left(-28\right)}}{2}$$

Whakareatia -4 ki te -28.

$$x=\frac{-3±\sqrt{9+112}}{2}$$

Tāpiri 9 ki te 112.

$$x=\frac{-3±\sqrt{121}}{2}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-3±11}{2}$$

Nā, me whakaoti te whārite x=\frac{-3±11}{2} ina he tāpiri te ±. Tāpiri -3 ki te 11.

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-3±11}{2} ina he tango te ±. Tango 11 mai i -3.

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

Whakawehe -14 ki te 2.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=4 x=-7$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-4.

$$x^{2}-4x+7\left(x-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$x^{2}-4x+7x-28=0$$

Pahekotia te -4x me 7x, ka 3x.

$$x^{2}+3x-28=0$$

Me tāpiri te 28 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}+3x=28$$

Whakawehea te 3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2}. Nā, tāpiria te pūrua o te \frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=28+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=28+\frac{9}{4}$$

Tāpiri 28 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{121}{4}$$

Tauwehea x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{11}{2} x+\frac{3}{2}=-\frac{11}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=4 x=-7$$