Whakaoti i te x^2-20x+36=0

$${ x }^{ 2 } -20x+36=0$$

Whakaoti mō x

$$x=2$$

$$x=18$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-20x+36 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.

$$a+b=-20 ab=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-18 b=-2$$

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

$$\left(x-18\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te x-2=0.

$$x=18 x=2$$

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+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-20 ab=1\times 36=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-18 b=-2$$

Tuhia anō te x^{2}-20x+36 hei \left(x^{2}-18x\right)+\left(-2x+36\right).

$$\left(x^{2}-18x\right)+\left(-2x+36\right)$$

Tauwehea te x i te tuatahi me te -2 i te rōpū tuarua.

$$x\left(x-18\right)-2\left(x-18\right)$$

Whakatauwehea atu te kīanga pātahi x-18 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-18\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te x-2=0.

$$x=18 x=2$$

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.

$$x^{2}-20x+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -20 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 36}}{2}$$

Pūrua -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 36}}{2}$$

Whakareatia -4 ki te 36.

$$x=\frac{-\left(-20\right)±\sqrt{400-144}}{2}$$

Tāpiri 400 ki te -144.

$$x=\frac{-\left(-20\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-20\right)±16}{2}$$

Ko te tauaro o -20 ko 20.

$$x=\frac{20±16}{2}$$

Nā, me whakaoti te whārite x=\frac{20±16}{2} ina he tāpiri te ±. Tāpiri 20 ki te 16.

$$x=\frac{36}{2}$$

Whakawehe 36 ki te 2.

$$x=18$$

Nā, me whakaoti te whārite x=\frac{20±16}{2} ina he tango te ±. Tango 16 mai i 20.

$$x=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=18 x=2$$

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.

$$x^{2}-20x+36=0$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$x^{2}-20x+36-36=-36$$

Mā te tango i te 36 i a ia ake anō ka toe ko te 0.

$$x^{2}-20x=-36$$

Whakawehea te -20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -10. Nā, tāpiria te pūrua o te -10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-20x+\left(-10\right)^{2}=-36+\left(-10\right)^{2}$$

Pūrua -10.

$$x^{2}-20x+100=-36+100$$

Tāpiri -36 ki te 100.

$$x^{2}-20x+100=64$$

Tauwehea x^{2}-20x+100. 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}.

$$\left(x-10\right)^{2}=64$$

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

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x-10=8 x-10=-8$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$x=18 x=2$$