Whakaoti i te 1/9x^2+x+9/4=0

$$\frac{ 1 }{ 9 } { x }^{ 2 } +x+ \frac{ 9 }{ 4 } =0$$

Whakaoti mō x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

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.

$$\frac{1}{9}x^{2}+x+\frac{9}{4}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{9} mō a, 1 mō b, me \frac{9}{4} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{9}\times \frac{9}{4}}}{2\times \frac{1}{9}}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times \frac{1}{9}\times \frac{9}{4}}}{2\times \frac{1}{9}}$$

Whakareatia -4 ki te \frac{1}{9}.

$$x=\frac{-1±\sqrt{1-\frac{4}{9}\times \frac{9}{4}}}{2\times \frac{1}{9}}$$

Whakareatia -\frac{4}{9} ki te \frac{9}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-1±\sqrt{1-1}}{2\times \frac{1}{9}}$$

Tāpiri 1 ki te -1.

$$x=\frac{-1±\sqrt{0}}{2\times \frac{1}{9}}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{1}{2\times \frac{1}{9}}$$

Whakareatia 2 ki te \frac{1}{9}.

$$x=-\frac{1}{\frac{2}{9}}$$

Whakawehe -1 ki te \frac{2}{9} mā te whakarea -1 ki te tau huripoki o \frac{2}{9}.

$$x=-\frac{9}{2}$$

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.

$$\frac{1}{9}x^{2}+x+\frac{9}{4}=0$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$\frac{1}{9}x^{2}+x+\frac{9}{4}-\frac{9}{4}=-\frac{9}{4}$$

Mā te tango i te \frac{9}{4} i a ia ake anō ka toe ko te 0.

$$\frac{1}{9}x^{2}+x=-\frac{9}{4}$$

Me whakarea ngā taha e rua ki te 9.

$$\frac{\frac{1}{9}x^{2}+x}{\frac{1}{9}}=-\frac{\frac{9}{4}}{\frac{1}{9}}$$

Mā te whakawehe ki te \frac{1}{9} ka wetekia te whakareanga ki te \frac{1}{9}.

$$x^{2}+\frac{1}{\frac{1}{9}}x=-\frac{\frac{9}{4}}{\frac{1}{9}}$$

Whakawehe 1 ki te \frac{1}{9} mā te whakarea 1 ki te tau huripoki o \frac{1}{9}.

$$x^{2}+9x=-\frac{\frac{9}{4}}{\frac{1}{9}}$$

Whakawehe -\frac{9}{4} ki te \frac{1}{9} mā te whakarea -\frac{9}{4} ki te tau huripoki o \frac{1}{9}.

$$x^{2}+9x=-\frac{81}{4}$$

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

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-\frac{81}{4}+\left(\frac{9}{2}\right)^{2}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+9x+\frac{81}{4}=\frac{-81+81}{4}$$

Tāpiri -\frac{81}{4} ki te \frac{81}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+9x+\frac{81}{4}=0$$

Tauwehea x^{2}+9x+\frac{81}{4}. 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}.

$$\left(x+\frac{9}{2}\right)^{2}=0$$

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

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+\frac{9}{2}=0 x+\frac{9}{2}=0$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$x=-\frac{9}{2} x=-\frac{9}{2}$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=-\frac{9}{2}$$