Whakaoti i te 1/8x^2-3/4x=2

$$\frac { 1 } { 8 } x ^ { 2 } - \frac { 3 } { 4 } x = 2$$

Whakaoti mō x

$$x=-2$$

$$x=8$$

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.

$$\frac{1}{8}x^{2}-\frac{3}{4}x=2$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$\frac{1}{8}x^{2}-\frac{3}{4}x-2=2-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$\frac{1}{8}x^{2}-\frac{3}{4}x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{8} mō a, -\frac{3}{4} mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\left(-\frac{3}{4}\right)^{2}-4\times \frac{1}{8}\left(-2\right)}}{2\times \frac{1}{8}}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-4\times \frac{1}{8}\left(-2\right)}}{2\times \frac{1}{8}}$$

Whakareatia -4 ki te \frac{1}{8}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-\frac{1}{2}\left(-2\right)}}{2\times \frac{1}{8}}$$

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

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+1}}{2\times \frac{1}{8}}$$

Tāpiri \frac{9}{16} ki te 1.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{25}{16}}}{2\times \frac{1}{8}}$$

Tuhia te pūtakerua o te \frac{25}{16}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\frac{5}{4}}{2\times \frac{1}{8}}$$

Ko te tauaro o -\frac{3}{4} ko \frac{3}{4}.

$$x=\frac{\frac{3}{4}±\frac{5}{4}}{2\times \frac{1}{8}}$$

Whakareatia 2 ki te \frac{1}{8}.

$$x=\frac{\frac{3}{4}±\frac{5}{4}}{\frac{1}{4}}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{4}±\frac{5}{4}}{\frac{1}{4}} ina he tāpiri te ±. Tāpiri \frac{3}{4} ki te \frac{5}{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.

$$x=\frac{2}{\frac{1}{4}}$$

Whakawehe 2 ki te \frac{1}{4} mā te whakarea 2 ki te tau huripoki o \frac{1}{4}.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{4}±\frac{5}{4}}{\frac{1}{4}} ina he tango te ±. Tango \frac{5}{4} mai i \frac{3}{4} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Whakawehe -\frac{1}{2} ki te \frac{1}{4} mā te whakarea -\frac{1}{2} ki te tau huripoki o \frac{1}{4}.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=8 x=-2$$

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.

$$\frac{1}{8}x^{2}-\frac{3}{4}x=2$$

Me whakarea ngā taha e rua ki te 8.

$$\frac{\frac{1}{8}x^{2}-\frac{3}{4}x}{\frac{1}{8}}=\frac{2}{\frac{1}{8}}$$

Mā te whakawehe ki te \frac{1}{8} ka wetekia te whakareanga ki te \frac{1}{8}.

$$x^{2}+\left(-\frac{\frac{3}{4}}{\frac{1}{8}}\right)x=\frac{2}{\frac{1}{8}}$$

Whakawehe -\frac{3}{4} ki te \frac{1}{8} mā te whakarea -\frac{3}{4} ki te tau huripoki o \frac{1}{8}.

$$x^{2}-6x=\frac{2}{\frac{1}{8}}$$

Whakawehe 2 ki te \frac{1}{8} mā te whakarea 2 ki te tau huripoki o \frac{1}{8}.

$$x^{2}-6x=16$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=16+9$$

Tāpiri 16 ki te 9.

$$x^{2}-6x+9=25$$

Tauwehea te x^{2}-6x+9. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=25$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-3=5 x-3=-5$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=8 x=-2$$