Whakaoti i te x^2-14x+65=25

$${ x }^{ 2 } -14x+65=25$$

Whakaoti mō x

$$x=4$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 25 mai i ngā taha e rua.

$$x^{2}-14x+65-25=0$$

Tangohia te 25 i te 65, ka 40.

$$x^{2}-14x+40=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-14x+40 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.

$$a+b=-14 ab=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-10 b=-4$$

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

$$\left(x-10\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x-4=0.

$$x=10 x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 25 mai i ngā taha e rua.

$$x^{2}-14x+65-25=0$$

Tangohia te 25 i te 65, ka 40.

$$x^{2}-14x+40=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=1\times 40=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-10 b=-4$$

Tuhia anō te x^{2}-14x+40 hei \left(x^{2}-10x\right)+\left(-4x+40\right).

$$\left(x^{2}-10x\right)+\left(-4x+40\right)$$

Tauwehea te x i te tuatahi me te -4 i te rōpū tuarua.

$$x\left(x-10\right)-4\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x-4=0.

$$x=10 x=4$$

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.

$$x^{2}-14x+65=25$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$x^{2}-14x+65-25=25-25$$

Mā te tango i te 25 i a ia ake anō ka toe ko te 0.

$$x^{2}-14x+65-25=0$$

Tango 25 mai i 65.

$$x^{2}-14x+40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me 40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 40}}{2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 40}}{2}$$

Whakareatia -4 ki te 40.

$$x=\frac{-\left(-14\right)±\sqrt{196-160}}{2}$$

Tāpiri 196 ki te -160.

$$x=\frac{-\left(-14\right)±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-14\right)±6}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±6}{2}$$

Nā, me whakaoti te whārite x=\frac{14±6}{2} ina he tāpiri te ±. Tāpiri 14 ki te 6.

$$x=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{14±6}{2} ina he tango te ±. Tango 6 mai i 14.

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=10 x=4$$

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.

$$x^{2}-14x+65=25$$

Me tango 65 mai i ngā taha e rua o te whārite.

$$x^{2}-14x+65-65=25-65$$

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

$$x^{2}-14x=25-65$$

Tango 65 mai i 25.

$$x^{2}-14x=-40$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-40+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-40+49$$

Tāpiri -40 ki te 49.

$$x^{2}-14x+49=9$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=9$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-7=3 x-7=-3$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=10 x=4$$