Whakaoti i te x^2-21x+104=0

$${ x }^{ 2 } -21x+104=0$$

Whakaoti mō x

$$x=8$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-21x+104 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.

$$a+b=-21 ab=104$$

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 104.

$$-1,-104 -2,-52 -4,-26 -8,-13$$

Tātaihia te tapeke mō ia takirua.

$$-1-104=-105 -2-52=-54 -4-26=-30 -8-13=-21$$

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

$$a=-13 b=-8$$

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

$$\left(x-13\right)\left(x-8\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x-8=0.

$$x=13 x=8$$

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+104. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-21 ab=1\times 104=104$$

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 104.

$$-1,-104 -2,-52 -4,-26 -8,-13$$

Tātaihia te tapeke mō ia takirua.

$$-1-104=-105 -2-52=-54 -4-26=-30 -8-13=-21$$

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

$$a=-13 b=-8$$

Tuhia anō te x^{2}-21x+104 hei \left(x^{2}-13x\right)+\left(-8x+104\right).

$$\left(x^{2}-13x\right)+\left(-8x+104\right)$$

Tauwehea te x i te tuatahi me te -8 i te rōpū tuarua.

$$x\left(x-13\right)-8\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(x-8\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x-8=0.

$$x=13 x=8$$

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.

$$x^{2}-21x+104=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -21 mō b, me 104 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 104}}{2}$$

Pūrua -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 104}}{2}$$

Whakareatia -4 ki te 104.

$$x=\frac{-\left(-21\right)±\sqrt{441-416}}{2}$$

Tāpiri 441 ki te -416.

$$x=\frac{-\left(-21\right)±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-21\right)±5}{2}$$

Ko te tauaro o -21 ko 21.

$$x=\frac{21±5}{2}$$

Nā, me whakaoti te whārite x=\frac{21±5}{2} ina he tāpiri te ±. Tāpiri 21 ki te 5.

$$x=\frac{26}{2}$$

Whakawehe 26 ki te 2.

$$x=13$$

Nā, me whakaoti te whārite x=\frac{21±5}{2} ina he tango te ±. Tango 5 mai i 21.

$$x=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$x=8$$

Kua oti te whārite te whakatau.

$$x=13 x=8$$

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.

$$x^{2}-21x+104=0$$

Me tango 104 mai i ngā taha e rua o te whārite.

$$x^{2}-21x+104-104=-104$$

Mā te tango i te 104 i a ia ake anō ka toe ko te 0.

$$x^{2}-21x=-104$$

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.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-104+\left(-\frac{21}{2}\right)^{2}$$

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

$$x^{2}-21x+\frac{441}{4}=-104+\frac{441}{4}$$

Tāpiri -104 ki te \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{25}{4}$$

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}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x-\frac{21}{2}=\frac{5}{2} x-\frac{21}{2}=-\frac{5}{2}$$

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

$$x=13 x=8$$