Whakaoti i te 9x^2+41x+9=0

$$9 { x }^{ 2 } +41x+9=0$$

Whakaoti mō x

$$x=\frac{\sqrt{1357}-41}{18}\approx -0.23125106$$

$$x=\frac{-\sqrt{1357}-41}{18}\approx -4.324304495$$

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.

$$9x^{2}+41x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 41 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-41±\sqrt{41^{2}-4\times 9\times 9}}{2\times 9}$$

Pūrua 41.

$$x=\frac{-41±\sqrt{1681-4\times 9\times 9}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-41±\sqrt{1681-36\times 9}}{2\times 9}$$

Whakareatia -36 ki te 9.

$$x=\frac{-41±\sqrt{1681-324}}{2\times 9}$$

Tāpiri 1681 ki te -324.

$$x=\frac{-41±\sqrt{1357}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-41±\sqrt{1357}}{18}$$

Nā, me whakaoti te whārite x=\frac{-41±\sqrt{1357}}{18} ina he tāpiri te ±. Tāpiri -41 ki te \sqrt{1357}.

$$x=\frac{\sqrt{1357}-41}{18}$$

Nā, me whakaoti te whārite x=\frac{-41±\sqrt{1357}}{18} ina he tango te ±. Tango \sqrt{1357} mai i -41.

$$x=\frac{-\sqrt{1357}-41}{18}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{1357}-41}{18} x=\frac{-\sqrt{1357}-41}{18}$$

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.

$$9x^{2}+41x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$9x^{2}+41x+9-9=-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$9x^{2}+41x=-9$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+41x}{9}=-\frac{9}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\frac{41}{9}x=-\frac{9}{9}$$

Whakawehe -9 ki te 9.

$$x^{2}+\frac{41}{9}x=-1$$

Whakawehea te \frac{41}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{41}{18}. Nā, tāpiria te pūrua o te \frac{41}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{41}{9}x+\left(\frac{41}{18}\right)^{2}=-1+\left(\frac{41}{18}\right)^{2}$$

Pūruatia \frac{41}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{41}{9}x+\frac{1681}{324}=-1+\frac{1681}{324}$$

Tāpiri -1 ki te \frac{1681}{324}.

$$x^{2}+\frac{41}{9}x+\frac{1681}{324}=\frac{1357}{324}$$

Tauwehea x^{2}+\frac{41}{9}x+\frac{1681}{324}. 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}.

$$\left(x+\frac{41}{18}\right)^{2}=\frac{1357}{324}$$

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

$$\sqrt{\left(x+\frac{41}{18}\right)^{2}}=\sqrt{\frac{1357}{324}}$$

Whakarūnātia.

$$x+\frac{41}{18}=\frac{\sqrt{1357}}{18} x+\frac{41}{18}=-\frac{\sqrt{1357}}{18}$$

Me tango \frac{41}{18} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1357}-41}{18} x=\frac{-\sqrt{1357}-41}{18}$$