Whakaoti i te -32x^2/130^2+x=264

$$\frac { - 32 x ^ { 2 } } { 130 ^ { 2 } } + x = 264$$

Whakaoti mō x (complex solution)

$$x=\frac{4225+65\sqrt{4223}i}{16}\approx 264.0625+263.999992602i$$

$$x=\frac{-65\sqrt{4223}i+4225}{16}\approx 264.0625-263.999992602i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 130 mā te pū o 2, kia riro ko 16900.

$$\frac{-32x^{2}}{16900}+x=264$$

Whakawehea te -32x^{2} ki te 16900, kia riro ko -\frac{8}{4225}x^{2}.

$$-\frac{8}{4225}x^{2}+x=264$$

Tangohia te 264 mai i ngā taha e rua.

$$-\frac{8}{4225}x^{2}+x-264=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{8}{4225} mō a, 1 mō b, me -264 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-\frac{8}{4225}\right)\left(-264\right)}}{2\left(-\frac{8}{4225}\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-\frac{8}{4225}\right)\left(-264\right)}}{2\left(-\frac{8}{4225}\right)}$$

Whakareatia -4 ki te -\frac{8}{4225}.

$$x=\frac{-1±\sqrt{1+\frac{32}{4225}\left(-264\right)}}{2\left(-\frac{8}{4225}\right)}$$

Whakareatia \frac{32}{4225} ki te -264.

$$x=\frac{-1±\sqrt{1-\frac{8448}{4225}}}{2\left(-\frac{8}{4225}\right)}$$

Tāpiri 1 ki te -\frac{8448}{4225}.

$$x=\frac{-1±\sqrt{-\frac{4223}{4225}}}{2\left(-\frac{8}{4225}\right)}$$

Tuhia te pūtakerua o te -\frac{4223}{4225}.

$$x=\frac{-1±\frac{\sqrt{4223}i}{65}}{2\left(-\frac{8}{4225}\right)}$$

Whakareatia 2 ki te -\frac{8}{4225}.

$$x=\frac{-1±\frac{\sqrt{4223}i}{65}}{-\frac{16}{4225}}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{\sqrt{4223}i}{65}}{-\frac{16}{4225}} ina he tāpiri te ±. Tāpiri -1 ki te \frac{i\sqrt{4223}}{65}.

$$x=\frac{\frac{\sqrt{4223}i}{65}-1}{-\frac{16}{4225}}$$

Whakawehe -1+\frac{i\sqrt{4223}}{65} ki te -\frac{16}{4225} mā te whakarea -1+\frac{i\sqrt{4223}}{65} ki te tau huripoki o -\frac{16}{4225}.

$$x=\frac{-65\sqrt{4223}i+4225}{16}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{\sqrt{4223}i}{65}}{-\frac{16}{4225}} ina he tango te ±. Tango \frac{i\sqrt{4223}}{65} mai i -1.

$$x=\frac{-\frac{\sqrt{4223}i}{65}-1}{-\frac{16}{4225}}$$

Whakawehe -1-\frac{i\sqrt{4223}}{65} ki te -\frac{16}{4225} mā te whakarea -1-\frac{i\sqrt{4223}}{65} ki te tau huripoki o -\frac{16}{4225}.

$$x=\frac{4225+65\sqrt{4223}i}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{-65\sqrt{4223}i+4225}{16} x=\frac{4225+65\sqrt{4223}i}{16}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 130 mā te pū o 2, kia riro ko 16900.

$$\frac{-32x^{2}}{16900}+x=264$$

Whakawehea te -32x^{2} ki te 16900, kia riro ko -\frac{8}{4225}x^{2}.

$$-\frac{8}{4225}x^{2}+x=264$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{4225}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{8}{4225}x^{2}+x}{-\frac{8}{4225}}=\frac{264}{-\frac{8}{4225}}$$

Mā te whakawehe ki te -\frac{8}{4225} ka wetekia te whakareanga ki te -\frac{8}{4225}.

$$x^{2}+\frac{1}{-\frac{8}{4225}}x=\frac{264}{-\frac{8}{4225}}$$

Whakawehe 1 ki te -\frac{8}{4225} mā te whakarea 1 ki te tau huripoki o -\frac{8}{4225}.

$$x^{2}-\frac{4225}{8}x=\frac{264}{-\frac{8}{4225}}$$

Whakawehe 264 ki te -\frac{8}{4225} mā te whakarea 264 ki te tau huripoki o -\frac{8}{4225}.

$$x^{2}-\frac{4225}{8}x=-139425$$

Whakawehea te -\frac{4225}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4225}{16}. Nā, tāpiria te pūrua o te -\frac{4225}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4225}{8}x+\left(-\frac{4225}{16}\right)^{2}=-139425+\left(-\frac{4225}{16}\right)^{2}$$

Pūruatia -\frac{4225}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4225}{8}x+\frac{17850625}{256}=-139425+\frac{17850625}{256}$$

Tāpiri -139425 ki te \frac{17850625}{256}.

$$x^{2}-\frac{4225}{8}x+\frac{17850625}{256}=-\frac{17842175}{256}$$

Tauwehea x^{2}-\frac{4225}{8}x+\frac{17850625}{256}. 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}.

$$\left(x-\frac{4225}{16}\right)^{2}=-\frac{17842175}{256}$$

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

$$\sqrt{\left(x-\frac{4225}{16}\right)^{2}}=\sqrt{-\frac{17842175}{256}}$$

Whakarūnātia.

$$x-\frac{4225}{16}=\frac{65\sqrt{4223}i}{16} x-\frac{4225}{16}=-\frac{65\sqrt{4223}i}{16}$$

Me tāpiri \frac{4225}{16} ki ngā taha e rua o te whārite.

$$x=\frac{4225+65\sqrt{4223}i}{16} x=\frac{-65\sqrt{4223}i+4225}{16}$$