Whakaoti i te x^2-18x=-65

$$x ^ { 2 } - 18 x = - 65$$

Whakaoti mō x

$$x=5$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 65 ki ngā taha e rua.

$$x^{2}-18x+65=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-18x+65 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.

$$a+b=-18 ab=65$$

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 65.

$$-1,-65 -5,-13$$

Tātaihia te tapeke mō ia takirua.

$$-1-65=-66 -5-13=-18$$

Ko te otinga te takirua ka hoatu i te tapeke -18.

$$a=-13 b=-5$$

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

$$\left(x-13\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x-5=0.

$$x=13 x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 65 ki ngā taha e rua.

$$x^{2}-18x+65=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+65. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-18 ab=1\times 65=65$$

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 65.

$$-1,-65 -5,-13$$

Tātaihia te tapeke mō ia takirua.

$$-1-65=-66 -5-13=-18$$

Ko te otinga te takirua ka hoatu i te tapeke -18.

$$a=-13 b=-5$$

Tuhia anō te x^{2}-18x+65 hei \left(x^{2}-13x\right)+\left(-5x+65\right).

$$\left(x^{2}-13x\right)+\left(-5x+65\right)$$

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

$$x\left(x-13\right)-5\left(x-13\right)$$

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

$$\left(x-13\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x-5=0.

$$x=13 x=5$$

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.

$$x^{2}-18x=-65$$

Me tāpiri 65 ki ngā taha e rua o te whārite.

$$x^{2}-18x-\left(-65\right)=-65-\left(-65\right)$$

Mā te tango i te -65 i a ia ake anō ka toe ko te 0.

$$x^{2}-18x-\left(-65\right)=0$$

Tango -65 mai i 0.

$$x^{2}-18x+65=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -18 mō b, me 65 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 65}}{2}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 65}}{2}$$

Whakareatia -4 ki te 65.

$$x=\frac{-\left(-18\right)±\sqrt{324-260}}{2}$$

Tāpiri 324 ki te -260.

$$x=\frac{-\left(-18\right)±\sqrt{64}}{2}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-\left(-18\right)±8}{2}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±8}{2}$$

Nā, me whakaoti te whārite x=\frac{18±8}{2} ina he tāpiri te ±. Tāpiri 18 ki te 8.

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

Whakawehe 26 ki te 2.

$$x=13$$

Nā, me whakaoti te whārite x=\frac{18±8}{2} ina he tango te ±. Tango 8 mai i 18.

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

Whakawehe 10 ki te 2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=13 x=5$$

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.

$$x^{2}-18x=-65$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-18x+\left(-9\right)^{2}=-65+\left(-9\right)^{2}$$

Pūrua -9.

$$x^{2}-18x+81=-65+81$$

Tāpiri -65 ki te 81.

$$x^{2}-18x+81=16$$

Tauwehea x^{2}-18x+81. 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}.

$$\left(x-9\right)^{2}=16$$

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

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$x-9=4 x-9=-4$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x=13 x=5$$