Whakaoti i te -y^2+10y+400=0

$$- { y }^{ 2 } +10y+400=0$$

Whakaoti mō y

$$y=5\sqrt{17}+5\approx 25.615528128$$

$$y=5-5\sqrt{17}\approx -15.615528128$$

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.

$$-y^{2}+10y+400=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 10 mō b, me 400 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\times 400}}{2\left(-1\right)}$$

Pūrua 10.

$$y=\frac{-10±\sqrt{100-4\left(-1\right)\times 400}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$y=\frac{-10±\sqrt{100+4\times 400}}{2\left(-1\right)}$$

Whakareatia 4 ki te 400.

$$y=\frac{-10±\sqrt{100+1600}}{2\left(-1\right)}$$

Tāpiri 100 ki te 1600.

$$y=\frac{-10±\sqrt{1700}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 1700.

$$y=\frac{-10±10\sqrt{17}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$y=\frac{-10±10\sqrt{17}}{-2}$$

Nā, me whakaoti te whārite y=\frac{-10±10\sqrt{17}}{-2} ina he tāpiri te ±. Tāpiri -10 ki te 10\sqrt{17}.

$$y=\frac{10\sqrt{17}-10}{-2}$$

Whakawehe -10+10\sqrt{17} ki te -2.

$$y=5-5\sqrt{17}$$

Nā, me whakaoti te whārite y=\frac{-10±10\sqrt{17}}{-2} ina he tango te ±. Tango 10\sqrt{17} mai i -10.

$$y=\frac{-10\sqrt{17}-10}{-2}$$

Whakawehe -10-10\sqrt{17} ki te -2.

$$y=5\sqrt{17}+5$$

Kua oti te whārite te whakatau.

$$y=5-5\sqrt{17} y=5\sqrt{17}+5$$

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.

$$-y^{2}+10y+400=0$$

Me tango 400 mai i ngā taha e rua o te whārite.

$$-y^{2}+10y+400-400=-400$$

Mā te tango i te 400 i a ia ake anō ka toe ko te 0.

$$-y^{2}+10y=-400$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-y^{2}+10y}{-1}=-\frac{400}{-1}$$

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

$$y^{2}+\frac{10}{-1}y=-\frac{400}{-1}$$

Whakawehe 10 ki te -1.

$$y^{2}-10y=-\frac{400}{-1}$$

Whakawehe -400 ki te -1.

$$y^{2}-10y=400$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-10y+\left(-5\right)^{2}=400+\left(-5\right)^{2}$$

Pūrua -5.

$$y^{2}-10y+25=400+25$$

Tāpiri 400 ki te 25.

$$y^{2}-10y+25=425$$

Tauwehea y^{2}-10y+25. 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}.

$$\left(y-5\right)^{2}=425$$

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

$$\sqrt{\left(y-5\right)^{2}}=\sqrt{425}$$

Whakarūnātia.

$$y-5=5\sqrt{17} y-5=-5\sqrt{17}$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$y=5\sqrt{17}+5 y=5-5\sqrt{17}$$