Whakaoti i te -x^2+8x+47=0

$$- x ^ { 2 } + 8 x + 47 = 0$$

Whakaoti mō x

$$x=3\sqrt{7}+4\approx 11.937253933$$

$$x=4-3\sqrt{7}\approx -3.937253933$$

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.

$$-x^{2}+8x+47=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 8 mō b, me 47 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\times 47}}{2\left(-1\right)}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\times 47}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-8±\sqrt{64+4\times 47}}{2\left(-1\right)}$$

Whakareatia 4 ki te 47.

$$x=\frac{-8±\sqrt{64+188}}{2\left(-1\right)}$$

Tāpiri 64 ki te 188.

$$x=\frac{-8±\sqrt{252}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 252.

$$x=\frac{-8±6\sqrt{7}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-8±6\sqrt{7}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-8±6\sqrt{7}}{-2} ina he tāpiri te ±. Tāpiri -8 ki te 6\sqrt{7}.

$$x=\frac{6\sqrt{7}-8}{-2}$$

Whakawehe -8+6\sqrt{7} ki te -2.

$$x=4-3\sqrt{7}$$

Nā, me whakaoti te whārite x=\frac{-8±6\sqrt{7}}{-2} ina he tango te ±. Tango 6\sqrt{7} mai i -8.

$$x=\frac{-6\sqrt{7}-8}{-2}$$

Whakawehe -8-6\sqrt{7} ki te -2.

$$x=3\sqrt{7}+4$$

Kua oti te whārite te whakatau.

$$x=4-3\sqrt{7} x=3\sqrt{7}+4$$

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.

$$-x^{2}+8x+47=0$$

Me tango 47 mai i ngā taha e rua o te whārite.

$$-x^{2}+8x+47-47=-47$$

Mā te tango i te 47 i a ia ake anō ka toe ko te 0.

$$-x^{2}+8x=-47$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+8x}{-1}=-\frac{47}{-1}$$

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

$$x^{2}+\frac{8}{-1}x=-\frac{47}{-1}$$

Whakawehe 8 ki te -1.

$$x^{2}-8x=-\frac{47}{-1}$$

Whakawehe -47 ki te -1.

$$x^{2}-8x=47$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=47+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=47+16$$

Tāpiri 47 ki te 16.

$$x^{2}-8x+16=63$$

Tauwehea x^{2}-8x+16. 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}.

$$\left(x-4\right)^{2}=63$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{63}$$

Whakarūnātia.

$$x-4=3\sqrt{7} x-4=-3\sqrt{7}$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=3\sqrt{7}+4 x=4-3\sqrt{7}$$