Whakaoti i te 32x^2+250x-1925=0

$$32 { x }^{ 2 } +250x-1925=0$$

Whakaoti mō x

$$x = \frac{5 \sqrt{3089} - 125}{32} \approx 4.77793327$$

$$x=\frac{-5\sqrt{3089}-125}{32}\approx -12.59043327$$

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.

$$32x^{2}+250x-1925=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 32 mō a, 250 mō b, me -1925 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-250±\sqrt{250^{2}-4\times 32\left(-1925\right)}}{2\times 32}$$

Pūrua 250.

$$x=\frac{-250±\sqrt{62500-4\times 32\left(-1925\right)}}{2\times 32}$$

Whakareatia -4 ki te 32.

$$x=\frac{-250±\sqrt{62500-128\left(-1925\right)}}{2\times 32}$$

Whakareatia -128 ki te -1925.

$$x=\frac{-250±\sqrt{62500+246400}}{2\times 32}$$

Tāpiri 62500 ki te 246400.

$$x=\frac{-250±\sqrt{308900}}{2\times 32}$$

Tuhia te pūtakerua o te 308900.

$$x=\frac{-250±10\sqrt{3089}}{2\times 32}$$

Whakareatia 2 ki te 32.

$$x=\frac{-250±10\sqrt{3089}}{64}$$

Nā, me whakaoti te whārite x=\frac{-250±10\sqrt{3089}}{64} ina he tāpiri te ±. Tāpiri -250 ki te 10\sqrt{3089}.

$$x=\frac{10\sqrt{3089}-250}{64}$$

Whakawehe -250+10\sqrt{3089} ki te 64.

$$x=\frac{5\sqrt{3089}-125}{32}$$

Nā, me whakaoti te whārite x=\frac{-250±10\sqrt{3089}}{64} ina he tango te ±. Tango 10\sqrt{3089} mai i -250.

$$x=\frac{-10\sqrt{3089}-250}{64}$$

Whakawehe -250-10\sqrt{3089} ki te 64.

$$x=\frac{-5\sqrt{3089}-125}{32}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{3089}-125}{32} x=\frac{-5\sqrt{3089}-125}{32}$$

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.

$$32x^{2}+250x-1925=0$$

Me tāpiri 1925 ki ngā taha e rua o te whārite.

$$32x^{2}+250x-1925-\left(-1925\right)=-\left(-1925\right)$$

Mā te tango i te -1925 i a ia ake anō ka toe ko te 0.

$$32x^{2}+250x=-\left(-1925\right)$$

Tango -1925 mai i 0.

$$32x^{2}+250x=1925$$

Whakawehea ngā taha e rua ki te 32.

$$\frac{32x^{2}+250x}{32}=\frac{1925}{32}$$

Mā te whakawehe ki te 32 ka wetekia te whakareanga ki te 32.

$$x^{2}+\frac{250}{32}x=\frac{1925}{32}$$

Whakahekea te hautanga \frac{250}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{125}{16}x=\frac{1925}{32}$$

Whakawehea te \frac{125}{16}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{125}{32}. Nā, tāpiria te pūrua o te \frac{125}{32} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{125}{16}x+\left(\frac{125}{32}\right)^{2}=\frac{1925}{32}+\left(\frac{125}{32}\right)^{2}$$

Pūruatia \frac{125}{32} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{125}{16}x+\frac{15625}{1024}=\frac{1925}{32}+\frac{15625}{1024}$$

Tāpiri \frac{1925}{32} ki te \frac{15625}{1024} 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.

$$x^{2}+\frac{125}{16}x+\frac{15625}{1024}=\frac{77225}{1024}$$

Tauwehea x^{2}+\frac{125}{16}x+\frac{15625}{1024}. 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}.

$$\left(x+\frac{125}{32}\right)^{2}=\frac{77225}{1024}$$

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

$$\sqrt{\left(x+\frac{125}{32}\right)^{2}}=\sqrt{\frac{77225}{1024}}$$

Whakarūnātia.

$$x+\frac{125}{32}=\frac{5\sqrt{3089}}{32} x+\frac{125}{32}=-\frac{5\sqrt{3089}}{32}$$

Me tango \frac{125}{32} mai i ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{3089}-125}{32} x=\frac{-5\sqrt{3089}-125}{32}$$