Whakaoti i te y^2-7y+6=0

$$y ^ { 2 } - 7 y + 6 = 0$$

Whakaoti mō y

$$y=1$$

$$y=6$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te y^{2}-7y+6 mā te whakamahi i te tātai y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=6$$

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 6.

$$-1,-6 -2,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-6=-7 -2-3=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-6 b=-1$$

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

$$\left(y-6\right)\left(y-1\right)$$

Hei kimi otinga whārite, me whakaoti te y-6=0 me te y-1=0.

$$y=6 y=1$$

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 y^{2}+ay+by+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=1\times 6=6$$

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 6.

$$-1,-6 -2,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-6=-7 -2-3=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-6 b=-1$$

Tuhia anō te y^{2}-7y+6 hei \left(y^{2}-6y\right)+\left(-y+6\right).

$$\left(y^{2}-6y\right)+\left(-y+6\right)$$

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

$$y\left(y-6\right)-\left(y-6\right)$$

Whakatauwehea atu te kīanga pātahi y-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-6\right)\left(y-1\right)$$

Hei kimi otinga whārite, me whakaoti te y-6=0 me te y-1=0.

$$y=6 y=1$$

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.

$$y^{2}-7y+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -7 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6}}{2}$$

Pūrua -7.

$$y=\frac{-\left(-7\right)±\sqrt{49-4\times 6}}{2}$$

Whakareatia -4 ki te 6.

$$y=\frac{-\left(-7\right)±\sqrt{49-24}}{2}$$

Tāpiri 49 ki te -24.

$$y=\frac{-\left(-7\right)±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$y=\frac{-\left(-7\right)±5}{2}$$

Ko te tauaro o -7 ko 7.

$$y=\frac{7±5}{2}$$

Nā, me whakaoti te whārite y=\frac{7±5}{2} ina he tāpiri te ±. Tāpiri 7 ki te 5.

$$y=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$y=6$$

Nā, me whakaoti te whārite y=\frac{7±5}{2} ina he tango te ±. Tango 5 mai i 7.

$$y=\frac{2}{2}$$

Whakawehe 2 ki te 2.

$$y=1$$

Kua oti te whārite te whakatau.

$$y=6 y=1$$

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.

$$y^{2}-7y+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$y^{2}-7y+6-6=-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$y^{2}-7y=-6$$

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

$$y^{2}-7y+\left(-\frac{7}{2}\right)^{2}=-6+\left(-\frac{7}{2}\right)^{2}$$

Pūruatia -\frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-7y+\frac{49}{4}=-6+\frac{49}{4}$$

Tāpiri -6 ki te \frac{49}{4}.

$$y^{2}-7y+\frac{49}{4}=\frac{25}{4}$$

Tauwehea y^{2}-7y+\frac{49}{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}.

$$\left(y-\frac{7}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(y-\frac{7}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$y-\frac{7}{2}=\frac{5}{2} y-\frac{7}{2}=-\frac{5}{2}$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$y=6 y=1$$