Whakaoti i te 2x-3*5/4x+3=1

$$2x-3 \times \frac{ 5 }{ 4x+3 } =1$$

Whakaoti mō x

$$x = \frac{\sqrt{145} - 1}{8} \approx 1.380199322$$

$$x=\frac{-\sqrt{145}-1}{8}\approx -1.630199322$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -\frac{3}{4} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 4x+3.

$$2x\left(4x+3\right)-3\times 5=4x+3$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 4x+3.

$$8x^{2}+6x-3\times 5=4x+3$$

Whakareatia te 3 ki te 5, ka 15.

$$8x^{2}+6x-15=4x+3$$

Tangohia te 4x mai i ngā taha e rua.

$$8x^{2}+6x-15-4x=3$$

Pahekotia te 6x me -4x, ka 2x.

$$8x^{2}+2x-15=3$$

Tangohia te 3 mai i ngā taha e rua.

$$8x^{2}+2x-15-3=0$$

Tangohia te 3 i te -15, ka -18.

$$8x^{2}+2x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 2 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 8\left(-18\right)}}{2\times 8}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times 8\left(-18\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-2±\sqrt{4-32\left(-18\right)}}{2\times 8}$$

Whakareatia -32 ki te -18.

$$x=\frac{-2±\sqrt{4+576}}{2\times 8}$$

Tāpiri 4 ki te 576.

$$x=\frac{-2±\sqrt{580}}{2\times 8}$$

Tuhia te pūtakerua o te 580.

$$x=\frac{-2±2\sqrt{145}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-2±2\sqrt{145}}{16}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{145}}{16} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{145}.

$$x=\frac{2\sqrt{145}-2}{16}$$

Whakawehe -2+2\sqrt{145} ki te 16.

$$x=\frac{\sqrt{145}-1}{8}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{145}}{16} ina he tango te ±. Tango 2\sqrt{145} mai i -2.

$$x=\frac{-2\sqrt{145}-2}{16}$$

Whakawehe -2-2\sqrt{145} ki te 16.

$$x=\frac{-\sqrt{145}-1}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{145}-1}{8} x=\frac{-\sqrt{145}-1}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -\frac{3}{4} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 4x+3.

$$2x\left(4x+3\right)-3\times 5=4x+3$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 4x+3.

$$8x^{2}+6x-3\times 5=4x+3$$

Whakareatia te 3 ki te 5, ka 15.

$$8x^{2}+6x-15=4x+3$$

Tangohia te 4x mai i ngā taha e rua.

$$8x^{2}+6x-15-4x=3$$

Pahekotia te 6x me -4x, ka 2x.

$$8x^{2}+2x-15=3$$

Me tāpiri te 15 ki ngā taha e rua.

$$8x^{2}+2x=3+15$$

Tāpirihia te 3 ki te 15, ka 18.

$$8x^{2}+2x=18$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+2x}{8}=\frac{18}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\frac{2}{8}x=\frac{18}{8}$$

Whakahekea te hautanga \frac{2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{1}{4}x=\frac{18}{8}$$

Whakahekea te hautanga \frac{18}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Whakawehea te \frac{1}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{8}. Nā, tāpiria te pūrua o te \frac{1}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=\frac{9}{4}+\left(\frac{1}{8}\right)^{2}$$

Pūruatia \frac{1}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{9}{4}+\frac{1}{64}$$

Tāpiri \frac{9}{4} ki te \frac{1}{64} 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.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{145}{64}$$

Tauwehea x^{2}+\frac{1}{4}x+\frac{1}{64}. 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}.

$$\left(x+\frac{1}{8}\right)^{2}=\frac{145}{64}$$

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

$$\sqrt{\left(x+\frac{1}{8}\right)^{2}}=\sqrt{\frac{145}{64}}$$

Whakarūnātia.

$$x+\frac{1}{8}=\frac{\sqrt{145}}{8} x+\frac{1}{8}=-\frac{\sqrt{145}}{8}$$

Me tango \frac{1}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{145}-1}{8} x=\frac{-\sqrt{145}-1}{8}$$