Whakaoti i te x^2-10x+21=0

$${ x }^{ 2 } -10x+21=0$$

Whakaoti mō x

$$x=3$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-10x+21 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.

$$a+b=-10 ab=21$$

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 21.

$$-1,-21 -3,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-21=-22 -3-7=-10$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-7 b=-3$$

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

$$\left(x-7\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-3=0.

$$x=7 x=3$$

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+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=1\times 21=21$$

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 21.

$$-1,-21 -3,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-21=-22 -3-7=-10$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-7 b=-3$$

Tuhia anō te x^{2}-10x+21 hei \left(x^{2}-7x\right)+\left(-3x+21\right).

$$\left(x^{2}-7x\right)+\left(-3x+21\right)$$

Tauwehea te x i te tuatahi me te -3 i te rōpū tuarua.

$$x\left(x-7\right)-3\left(x-7\right)$$

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

$$\left(x-7\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-3=0.

$$x=7 x=3$$

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.

$$x^{2}-10x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 21}}{2}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 21}}{2}$$

Whakareatia -4 ki te 21.

$$x=\frac{-\left(-10\right)±\sqrt{100-84}}{2}$$

Tāpiri 100 ki te -84.

$$x=\frac{-\left(-10\right)±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-\left(-10\right)±4}{2}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±4}{2}$$

Nā, me whakaoti te whārite x=\frac{10±4}{2} ina he tāpiri te ±. Tāpiri 10 ki te 4.

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

Whakawehe 14 ki te 2.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{10±4}{2} ina he tango te ±. Tango 4 mai i 10.

$$x=\frac{6}{2}$$

Whakawehe 6 ki te 2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=7 x=3$$

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.

$$x^{2}-10x+21=0$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$x^{2}-10x+21-21=-21$$

Mā te tango i te 21 i a ia ake anō ka toe ko te 0.

$$x^{2}-10x=-21$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-10x+\left(-5\right)^{2}=-21+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=-21+25$$

Tāpiri -21 ki te 25.

$$x^{2}-10x+25=4$$

Tauwehea x^{2}-10x+25. 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}.

$$\left(x-5\right)^{2}=4$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-5=2 x-5=-2$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$x=7 x=3$$