Whakaoti i te 25x^2-8x=12x-4

$$25 { x }^{ 2 } -8x=12x-4$$

Whakaoti mō x

$$x=\frac{2}{5}=0.4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 12x mai i ngā taha e rua.

$$25x^{2}-8x-12x=-4$$

Pahekotia te -8x me -12x, ka -20x.

$$25x^{2}-20x=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$25x^{2}-20x+4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 25x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-20 ab=25\times 4=100$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-10 b=-10$$

Tuhia anō te 25x^{2}-20x+4 hei \left(25x^{2}-10x\right)+\left(-10x+4\right).

$$\left(25x^{2}-10x\right)+\left(-10x+4\right)$$

Tauwehea te 5x i te tuatahi me te -2 i te rōpū tuarua.

$$5x\left(5x-2\right)-2\left(5x-2\right)$$

Whakatauwehea atu te kīanga pātahi 5x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-2\right)\left(5x-2\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(5x-2\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te 5x-2=0.

$$x=\frac{2}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 12x mai i ngā taha e rua.

$$25x^{2}-8x-12x=-4$$

Pahekotia te -8x me -12x, ka -20x.

$$25x^{2}-20x=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$25x^{2}-20x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, -20 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 25\times 4}}{2\times 25}$$

Pūrua -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 25\times 4}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-\left(-20\right)±\sqrt{400-100\times 4}}{2\times 25}$$

Whakareatia -100 ki te 4.

$$x=\frac{-\left(-20\right)±\sqrt{400-400}}{2\times 25}$$

Tāpiri 400 ki te -400.

$$x=\frac{-\left(-20\right)±\sqrt{0}}{2\times 25}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-20}{2\times 25}$$

Ko te tauaro o -20 ko 20.

$$x=\frac{20}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{20}{50}$$

Whakahekea te hautanga \frac{20}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{2}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 12x mai i ngā taha e rua.

$$25x^{2}-8x-12x=-4$$

Pahekotia te -8x me -12x, ka -20x.

$$25x^{2}-20x=-4$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}-20x}{25}=-\frac{4}{25}$$

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

$$x^{2}+\left(-\frac{20}{25}\right)x=-\frac{4}{25}$$

Whakahekea te hautanga \frac{-20}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{4}{5}x=-\frac{4}{25}$$

Whakawehea te -\frac{4}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{5}. Nā, tāpiria te pūrua o te -\frac{2}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4}{5}x+\left(-\frac{2}{5}\right)^{2}=-\frac{4}{25}+\left(-\frac{2}{5}\right)^{2}$$

Pūruatia -\frac{2}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=\frac{-4+4}{25}$$

Tāpiri -\frac{4}{25} ki te \frac{4}{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.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=0$$

Tauwehea te x^{2}-\frac{4}{5}x+\frac{4}{25}. 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}.

$$\left(x-\frac{2}{5}\right)^{2}=0$$

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

$$\sqrt{\left(x-\frac{2}{5}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-\frac{2}{5}=0 x-\frac{2}{5}=0$$

Me tāpiri \frac{2}{5} ki ngā taha e rua o te whārite.

$$x=\frac{2}{5} x=\frac{2}{5}$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=\frac{2}{5}$$