Whakaoti i te x^2-12x=-35

$$x ^ { 2 } - 12 x = - 35$$

Whakaoti mō x

$$x=5$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 35 ki ngā taha e rua.

$$x^{2}-12x+35=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-12x+35 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.

$$a+b=-12 ab=35$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 35.

$$-1,-35 -5,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-35=-36 -5-7=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-7 b=-5$$

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

$$\left(x-7\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-5=0.

$$x=7 x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 35 ki ngā taha e rua.

$$x^{2}-12x+35=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=1\times 35=35$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 35.

$$-1,-35 -5,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-35=-36 -5-7=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-7 b=-5$$

Tuhia anō te x^{2}-12x+35 hei \left(x^{2}-7x\right)+\left(-5x+35\right).

$$\left(x^{2}-7x\right)+\left(-5x+35\right)$$

Tauwehea te x i te tuatahi me te -5 i te rōpū tuarua.

$$x\left(x-7\right)-5\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-5=0.

$$x=7 x=5$$

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.

$$x^{2}-12x=-35$$

Me tāpiri 35 ki ngā taha e rua o te whārite.

$$x^{2}-12x-\left(-35\right)=-35-\left(-35\right)$$

Mā te tango i te -35 i a ia ake anō ka toe ko te 0.

$$x^{2}-12x-\left(-35\right)=0$$

Tango -35 mai i 0.

$$x^{2}-12x+35=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -12 mō b, me 35 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 35}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 35}}{2}$$

Whakareatia -4 ki te 35.

$$x=\frac{-\left(-12\right)±\sqrt{144-140}}{2}$$

Tāpiri 144 ki te -140.

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

Tuhia te pūtakerua o te 4.

$$x=\frac{-\left(-12\right)±2}{2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±2}{2}$$

Nā, me whakaoti te whārite x=\frac{12±2}{2} ina he tāpiri te ±. Tāpiri 12 ki te 2.

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

Whakawehe 14 ki te 2.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{12±2}{2} ina he tango te ±. Tango 2 mai i 12.

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

Whakawehe 10 ki te 2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=7 x=5$$

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.

$$x^{2}-12x=-35$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=-35+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=-35+36$$

Tāpiri -35 ki te 36.

$$x^{2}-12x+36=1$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=1$$

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

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-6=1 x-6=-1$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=7 x=5$$