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

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

Whakaoti mō y

$$y=-1$$

$$y=7$$

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+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=7 b=-1$$

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

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

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

$$-y\left(y-7\right)-\left(y-7\right)$$

Whakatauwehea atu te kīanga pātahi y-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-7\right)\left(-y-1\right)$$

Hei kimi otinga whārite, me whakaoti te y-7=0 me te -y-1=0.

$$y=7 y=-1$$

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.

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

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

$$y=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Pūrua 6.

$$y=\frac{-6±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$y=\frac{-6±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Whakareatia 4 ki te 7.

$$y=\frac{-6±\sqrt{36+28}}{2\left(-1\right)}$$

Tāpiri 36 ki te 28.

$$y=\frac{-6±\sqrt{64}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 64.

$$y=\frac{-6±8}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$y=\frac{-6±8}{-2}$$

Nā, me whakaoti te whārite y=\frac{-6±8}{-2} ina he tāpiri te ±. Tāpiri -6 ki te 8.

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

Whakawehe 2 ki te -2.

$$y=-1$$

Nā, me whakaoti te whārite y=\frac{-6±8}{-2} ina he tango te ±. Tango 8 mai i -6.

$$y=-\frac{14}{-2}$$

Whakawehe -14 ki te -2.

$$y=7$$

Kua oti te whārite te whakatau.

$$y=-1 y=7$$

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.

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

Me tango 7 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-y^{2}+6y}{-1}=-\frac{7}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$y^{2}+\frac{6}{-1}y=-\frac{7}{-1}$$

Whakawehe 6 ki te -1.

$$y^{2}-6y=-\frac{7}{-1}$$

Whakawehe -7 ki te -1.

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

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-6y+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Pūrua -3.

$$y^{2}-6y+9=7+9$$

Tāpiri 7 ki te 9.

$$y^{2}-6y+9=16$$

Tauwehea y^{2}-6y+9. 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}.

$$\left(y-3\right)^{2}=16$$

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

$$\sqrt{\left(y-3\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$y-3=4 y-3=-4$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$y=7 y=-1$$