Whakaoti i te 4:2/x-4/5=3/x

$$4 : \frac { 2 } { x } - \frac { 4 } { 5 } = \frac { 3 } { x }$$

Whakaoti mō x

$$x=-\frac{\sqrt{154}}{10}+\frac{1}{5}\approx -1.040967365$$

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5}\approx 1.440967365$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5x, arā, te tauraro pātahi he tino iti rawa te kitea o 5,x.

$$\frac{5}{2}x^{2}\times 4+5x\left(-\frac{4}{5}\right)=5\times 3$$

Whakareatia te \frac{5}{2} ki te 4, ka 10.

$$10x^{2}+5x\left(-\frac{4}{5}\right)=5\times 3$$

Whakareatia te 5 ki te -\frac{4}{5}, ka -4.

$$10x^{2}-4x=5\times 3$$

Whakareatia te 5 ki te 3, ka 15.

$$10x^{2}-4x=15$$

Tangohia te 15 mai i ngā taha e rua.

$$10x^{2}-4x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, -4 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 10\left(-15\right)}}{2\times 10}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 10\left(-15\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-4\right)±\sqrt{16-40\left(-15\right)}}{2\times 10}$$

Whakareatia -40 ki te -15.

$$x=\frac{-\left(-4\right)±\sqrt{16+600}}{2\times 10}$$

Tāpiri 16 ki te 600.

$$x=\frac{-\left(-4\right)±\sqrt{616}}{2\times 10}$$

Tuhia te pūtakerua o te 616.

$$x=\frac{-\left(-4\right)±2\sqrt{154}}{2\times 10}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{154}}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{4±2\sqrt{154}}{20}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{154}}{20} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{154}.

$$x=\frac{2\sqrt{154}+4}{20}$$

Whakawehe 4+2\sqrt{154} ki te 20.

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{154}}{20} ina he tango te ±. Tango 2\sqrt{154} mai i 4.

$$x=\frac{4-2\sqrt{154}}{20}$$

Whakawehe 4-2\sqrt{154} ki te 20.

$$x=-\frac{\sqrt{154}}{10}+\frac{1}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5} x=-\frac{\sqrt{154}}{10}+\frac{1}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5x, arā, te tauraro pātahi he tino iti rawa te kitea o 5,x.

$$\frac{5}{2}x^{2}\times 4+5x\left(-\frac{4}{5}\right)=5\times 3$$

Whakareatia te \frac{5}{2} ki te 4, ka 10.

$$10x^{2}+5x\left(-\frac{4}{5}\right)=5\times 3$$

Whakareatia te 5 ki te -\frac{4}{5}, ka -4.

$$10x^{2}-4x=5\times 3$$

Whakareatia te 5 ki te 3, ka 15.

$$10x^{2}-4x=15$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}-4x}{10}=\frac{15}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}+\left(-\frac{4}{10}\right)x=\frac{15}{10}$$

Whakahekea te hautanga \frac{-4}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{2}{5}x=\frac{15}{10}$$

Whakahekea te hautanga \frac{15}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

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

Whakawehea te -\frac{2}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{5}. Nā, tāpiria te pūrua o te -\frac{1}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{3}{2}+\frac{1}{25}$$

Tāpiri \frac{3}{2} ki te \frac{1}{25} 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.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{77}{50}$$

Tauwehea x^{2}-\frac{2}{5}x+\frac{1}{25}. 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}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{77}{50}$$

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

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{77}{50}}$$

Whakarūnātia.

$$x-\frac{1}{5}=\frac{\sqrt{154}}{10} x-\frac{1}{5}=-\frac{\sqrt{154}}{10}$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5} x=-\frac{\sqrt{154}}{10}+\frac{1}{5}$$