Whakaoti i te x^2-21x+20=0

$$x ^ { 2 } - 21 x + 20 = 0$$

Whakaoti mō x

$$x=1$$

$$x=20$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-21x+20 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.

$$a+b=-21 ab=20$$

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 20.

$$-1,-20 -2,-10 -4,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

Ko te otinga te takirua ka hoatu i te tapeke -21.

$$a=-20 b=-1$$

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

$$\left(x-20\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x-1=0.

$$x=20 x=1$$

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+20. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-21 ab=1\times 20=20$$

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 20.

$$-1,-20 -2,-10 -4,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

Ko te otinga te takirua ka hoatu i te tapeke -21.

$$a=-20 b=-1$$

Tuhia anō te x^{2}-21x+20 hei \left(x^{2}-20x\right)+\left(-x+20\right).

$$\left(x^{2}-20x\right)+\left(-x+20\right)$$

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

$$x\left(x-20\right)-\left(x-20\right)$$

Whakatauwehea atu te kīanga pātahi x-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-20\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x-1=0.

$$x=20 x=1$$

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.

$$x^{2}-21x+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -21 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 20}}{2}$$

Pūrua -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 20}}{2}$$

Whakareatia -4 ki te 20.

$$x=\frac{-\left(-21\right)±\sqrt{441-80}}{2}$$

Tāpiri 441 ki te -80.

$$x=\frac{-\left(-21\right)±\sqrt{361}}{2}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-\left(-21\right)±19}{2}$$

Ko te tauaro o -21 ko 21.

$$x=\frac{21±19}{2}$$

Nā, me whakaoti te whārite x=\frac{21±19}{2} ina he tāpiri te ±. Tāpiri 21 ki te 19.

$$x=\frac{40}{2}$$

Whakawehe 40 ki te 2.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{21±19}{2} ina he tango te ±. Tango 19 mai i 21.

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

Whakawehe 2 ki te 2.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=20 x=1$$

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.

$$x^{2}-21x+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$x^{2}-21x+20-20=-20$$

Mā te tango i te 20 i a ia ake anō ka toe ko te 0.

$$x^{2}-21x=-20$$

Whakawehea te -21, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{21}{2}. Nā, tāpiria te pūrua o te -\frac{21}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-20+\left(-\frac{21}{2}\right)^{2}$$

Pūruatia -\frac{21}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-21x+\frac{441}{4}=-20+\frac{441}{4}$$

Tāpiri -20 ki te \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{361}{4}$$

Tauwehea x^{2}-21x+\frac{441}{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}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{361}{4}$$

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

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Whakarūnātia.

$$x-\frac{21}{2}=\frac{19}{2} x-\frac{21}{2}=-\frac{19}{2}$$

Me tāpiri \frac{21}{2} ki ngā taha e rua o te whārite.

$$x=20 x=1$$