Whakaoti i te -1/5x^2+3x+16/5=0

$$- \frac{ 1 }{ 5 } { x }^{ 2 } +3x+ \frac{ 16 }{ 5 } =0$$

Whakaoti mō x

$$x=-1$$

$$x=16$$

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.

$$-\frac{1}{5}x^{2}+3x+\frac{16}{5}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{5} mō a, 3 mō b, me \frac{16}{5} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-\frac{1}{5}\right)\times \frac{16}{5}}}{2\left(-\frac{1}{5}\right)}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\left(-\frac{1}{5}\right)\times \frac{16}{5}}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia -4 ki te -\frac{1}{5}.

$$x=\frac{-3±\sqrt{9+\frac{4}{5}\times \frac{16}{5}}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia \frac{4}{5} ki te \frac{16}{5} 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.

$$x=\frac{-3±\sqrt{9+\frac{64}{25}}}{2\left(-\frac{1}{5}\right)}$$

Tāpiri 9 ki te \frac{64}{25}.

$$x=\frac{-3±\sqrt{\frac{289}{25}}}{2\left(-\frac{1}{5}\right)}$$

Tuhia te pūtakerua o te \frac{289}{25}.

$$x=\frac{-3±\frac{17}{5}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia 2 ki te -\frac{1}{5}.

$$x=\frac{-3±\frac{17}{5}}{-\frac{2}{5}}$$

Nā, me whakaoti te whārite x=\frac{-3±\frac{17}{5}}{-\frac{2}{5}} ina he tāpiri te ±. Tāpiri -3 ki te \frac{17}{5}.

$$x=\frac{\frac{2}{5}}{-\frac{2}{5}}$$

Whakawehe \frac{2}{5} ki te -\frac{2}{5} mā te whakarea \frac{2}{5} ki te tau huripoki o -\frac{2}{5}.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-3±\frac{17}{5}}{-\frac{2}{5}} ina he tango te ±. Tango \frac{17}{5} mai i -3.

$$x=-\frac{\frac{32}{5}}{-\frac{2}{5}}$$

Whakawehe -\frac{32}{5} ki te -\frac{2}{5} mā te whakarea -\frac{32}{5} ki te tau huripoki o -\frac{2}{5}.

$$x=16$$

Kua oti te whārite te whakatau.

$$x=-1 x=16$$

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.

$$-\frac{1}{5}x^{2}+3x+\frac{16}{5}=0$$

Me tango \frac{16}{5} mai i ngā taha e rua o te whārite.

$$-\frac{1}{5}x^{2}+3x+\frac{16}{5}-\frac{16}{5}=-\frac{16}{5}$$

Mā te tango i te \frac{16}{5} i a ia ake anō ka toe ko te 0.

$$-\frac{1}{5}x^{2}+3x=-\frac{16}{5}$$

Me whakarea ngā taha e rua ki te -5.

$$\frac{-\frac{1}{5}x^{2}+3x}{-\frac{1}{5}}=-\frac{\frac{16}{5}}{-\frac{1}{5}}$$

Mā te whakawehe ki te -\frac{1}{5} ka wetekia te whakareanga ki te -\frac{1}{5}.

$$x^{2}+\frac{3}{-\frac{1}{5}}x=-\frac{\frac{16}{5}}{-\frac{1}{5}}$$

Whakawehe 3 ki te -\frac{1}{5} mā te whakarea 3 ki te tau huripoki o -\frac{1}{5}.

$$x^{2}-15x=-\frac{\frac{16}{5}}{-\frac{1}{5}}$$

Whakawehe -\frac{16}{5} ki te -\frac{1}{5} mā te whakarea -\frac{16}{5} ki te tau huripoki o -\frac{1}{5}.

$$x^{2}-15x=16$$

Whakawehea te -15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{2}. Nā, tāpiria te pūrua o te -\frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=16+\left(-\frac{15}{2}\right)^{2}$$

Pūruatia -\frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-15x+\frac{225}{4}=16+\frac{225}{4}$$

Tāpiri 16 ki te \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{289}{4}$$

Tauwehea x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{289}{4}$$

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

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Whakarūnātia.

$$x-\frac{15}{2}=\frac{17}{2} x-\frac{15}{2}=-\frac{17}{2}$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$x=16 x=-1$$