Whakaoti i te y^2-10y+13=0

$${ y }^{ 2 } -10y+13=0$$

Whakaoti mō y

$$y=2\sqrt{3}+5\approx 8.464101615$$

$$y=5-2\sqrt{3}\approx 1.535898385$$

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.

$$y^{2}-10y+13=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me 13 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 13}}{2}$$

Pūrua -10.

$$y=\frac{-\left(-10\right)±\sqrt{100-4\times 13}}{2}$$

Whakareatia -4 ki te 13.

$$y=\frac{-\left(-10\right)±\sqrt{100-52}}{2}$$

Tāpiri 100 ki te -52.

$$y=\frac{-\left(-10\right)±\sqrt{48}}{2}$$

Tuhia te pūtakerua o te 48.

$$y=\frac{-\left(-10\right)±4\sqrt{3}}{2}$$

Ko te tauaro o -10 ko 10.

$$y=\frac{10±4\sqrt{3}}{2}$$

Nā, me whakaoti te whārite y=\frac{10±4\sqrt{3}}{2} ina he tāpiri te ±. Tāpiri 10 ki te 4\sqrt{3}.

$$y=\frac{4\sqrt{3}+10}{2}$$

Whakawehe 10+4\sqrt{3} ki te 2.

$$y=2\sqrt{3}+5$$

Nā, me whakaoti te whārite y=\frac{10±4\sqrt{3}}{2} ina he tango te ±. Tango 4\sqrt{3} mai i 10.

$$y=\frac{10-4\sqrt{3}}{2}$$

Whakawehe 10-4\sqrt{3} ki te 2.

$$y=5-2\sqrt{3}$$

Kua oti te whārite te whakatau.

$$y=2\sqrt{3}+5 y=5-2\sqrt{3}$$

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.

$$y^{2}-10y+13=0$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$y^{2}-10y+13-13=-13$$

Mā te tango i te 13 i a ia ake anō ka toe ko te 0.

$$y^{2}-10y=-13$$

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.

$$y^{2}-10y+\left(-5\right)^{2}=-13+\left(-5\right)^{2}$$

Pūrua -5.

$$y^{2}-10y+25=-13+25$$

Tāpiri -13 ki te 25.

$$y^{2}-10y+25=12$$

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}.

$$\left(y-5\right)^{2}=12$$

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

$$\sqrt{\left(y-5\right)^{2}}=\sqrt{12}$$

Whakarūnātia.

$$y-5=2\sqrt{3} y-5=-2\sqrt{3}$$

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

$$y=2\sqrt{3}+5 y=5-2\sqrt{3}$$