Whakaoti i te 4y^2-56y=108

$$4 y ^ { 2 } - 56 y = 108$$

Whakaoti mō y

$$y=2\sqrt{19}+7\approx 15.717797887$$

$$y=7-2\sqrt{19}\approx -1.717797887$$

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.

$$4y^{2}-56y=108$$

Me tango 108 mai i ngā taha e rua o te whārite.

$$4y^{2}-56y-108=108-108$$

Mā te tango i te 108 i a ia ake anō ka toe ko te 0.

$$4y^{2}-56y-108=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -56 mō b, me -108 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 4\left(-108\right)}}{2\times 4}$$

Pūrua -56.

$$y=\frac{-\left(-56\right)±\sqrt{3136-4\times 4\left(-108\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$y=\frac{-\left(-56\right)±\sqrt{3136-16\left(-108\right)}}{2\times 4}$$

Whakareatia -16 ki te -108.

$$y=\frac{-\left(-56\right)±\sqrt{3136+1728}}{2\times 4}$$

Tāpiri 3136 ki te 1728.

$$y=\frac{-\left(-56\right)±\sqrt{4864}}{2\times 4}$$

Tuhia te pūtakerua o te 4864.

$$y=\frac{-\left(-56\right)±16\sqrt{19}}{2\times 4}$$

Ko te tauaro o -56 ko 56.

$$y=\frac{56±16\sqrt{19}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{56±16\sqrt{19}}{8}$$

Nā, me whakaoti te whārite y=\frac{56±16\sqrt{19}}{8} ina he tāpiri te ±. Tāpiri 56 ki te 16\sqrt{19}.

$$y=\frac{16\sqrt{19}+56}{8}$$

Whakawehe 56+16\sqrt{19} ki te 8.

$$y=2\sqrt{19}+7$$

Nā, me whakaoti te whārite y=\frac{56±16\sqrt{19}}{8} ina he tango te ±. Tango 16\sqrt{19} mai i 56.

$$y=\frac{56-16\sqrt{19}}{8}$$

Whakawehe 56-16\sqrt{19} ki te 8.

$$y=7-2\sqrt{19}$$

Kua oti te whārite te whakatau.

$$y=2\sqrt{19}+7 y=7-2\sqrt{19}$$

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.

$$4y^{2}-56y=108$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4y^{2}-56y}{4}=\frac{108}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$y^{2}+\left(-\frac{56}{4}\right)y=\frac{108}{4}$$

Whakawehe -56 ki te 4.

$$y^{2}-14y=\frac{108}{4}$$

Whakawehe 108 ki te 4.

$$y^{2}-14y=27$$

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.

$$y^{2}-14y+\left(-7\right)^{2}=27+\left(-7\right)^{2}$$

Pūrua -7.

$$y^{2}-14y+49=27+49$$

Tāpiri 27 ki te 49.

$$y^{2}-14y+49=76$$

Tauwehea y^{2}-14y+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}.

$$\left(y-7\right)^{2}=76$$

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

$$\sqrt{\left(y-7\right)^{2}}=\sqrt{76}$$

Whakarūnātia.

$$y-7=2\sqrt{19} y-7=-2\sqrt{19}$$

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

$$y=2\sqrt{19}+7 y=7-2\sqrt{19}$$