Whakaoti i te (2x-3)(4x-2)-x(2x-3)=0

$$( 2 x - 3 ) ( 4 x - 2 ) - x ( 2 x - 3 ) = 0$$

Whakaoti mō x

$$x=\frac{2}{3}\approx 0.666666667$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-3 ki te 4x-2 ka whakakotahi i ngā kupu rite.

$$8x^{2}-16x+6-x\left(2x-3\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2x-3.

$$8x^{2}-16x+6-\left(2x^{2}-3x\right)=0$$

Hei kimi i te tauaro o 2x^{2}-3x, kimihia te tauaro o ia taurangi.

$$8x^{2}-16x+6-2x^{2}+3x=0$$

Pahekotia te 8x^{2} me -2x^{2}, ka 6x^{2}.

$$6x^{2}-16x+6+3x=0$$

Pahekotia te -16x me 3x, ka -13x.

$$6x^{2}-13x+6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6x^{2}+ax+bx+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-13 ab=6\times 6=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=-9 b=-4$$

Tuhia anō te 6x^{2}-13x+6 hei \left(6x^{2}-9x\right)+\left(-4x+6\right).

$$\left(6x^{2}-9x\right)+\left(-4x+6\right)$$

Tauwehea te 3x i te tuatahi me te -2 i te rōpū tuarua.

$$3x\left(2x-3\right)-2\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(3x-2\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-3=0 me te 3x-2=0.

$$x=\frac{3}{2} x=\frac{2}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-3 ki te 4x-2 ka whakakotahi i ngā kupu rite.

$$8x^{2}-16x+6-x\left(2x-3\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2x-3.

$$8x^{2}-16x+6-\left(2x^{2}-3x\right)=0$$

Hei kimi i te tauaro o 2x^{2}-3x, kimihia te tauaro o ia taurangi.

$$8x^{2}-16x+6-2x^{2}+3x=0$$

Pahekotia te 8x^{2} me -2x^{2}, ka 6x^{2}.

$$6x^{2}-16x+6+3x=0$$

Pahekotia te -16x me 3x, ka -13x.

$$6x^{2}-13x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -13 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\times 6}}{2\times 6}$$

Pūrua -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 6\times 6}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-13\right)±\sqrt{169-24\times 6}}{2\times 6}$$

Whakareatia -24 ki te 6.

$$x=\frac{-\left(-13\right)±\sqrt{169-144}}{2\times 6}$$

Tāpiri 169 ki te -144.

$$x=\frac{-\left(-13\right)±\sqrt{25}}{2\times 6}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-13\right)±5}{2\times 6}$$

Ko te tauaro o -13 ko 13.

$$x=\frac{13±5}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{13±5}{12}$$

Nā, me whakaoti te whārite x=\frac{13±5}{12} ina he tāpiri te ±. Tāpiri 13 ki te 5.

$$x=\frac{18}{12}$$

Whakahekea te hautanga \frac{18}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{13±5}{12} ina he tango te ±. Tango 5 mai i 13.

$$x=\frac{8}{12}$$

Whakahekea te hautanga \frac{8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{2}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{3}{2} x=\frac{2}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-3 ki te 4x-2 ka whakakotahi i ngā kupu rite.

$$8x^{2}-16x+6-x\left(2x-3\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2x-3.

$$8x^{2}-16x+6-\left(2x^{2}-3x\right)=0$$

Hei kimi i te tauaro o 2x^{2}-3x, kimihia te tauaro o ia taurangi.

$$8x^{2}-16x+6-2x^{2}+3x=0$$

Pahekotia te 8x^{2} me -2x^{2}, ka 6x^{2}.

$$6x^{2}-16x+6+3x=0$$

Pahekotia te -16x me 3x, ka -13x.

$$6x^{2}-13x+6=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$6x^{2}-13x=-6$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-13x}{6}=-\frac{6}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}-\frac{13}{6}x=-\frac{6}{6}$$

Whakawehe -6 ki te 6.

$$x^{2}-\frac{13}{6}x=-1$$

Whakawehea te -\frac{13}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{12}. Nā, tāpiria te pūrua o te -\frac{13}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=-1+\left(-\frac{13}{12}\right)^{2}$$

Pūruatia -\frac{13}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-1+\frac{169}{144}$$

Tāpiri -1 ki te \frac{169}{144}.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{25}{144}$$

Tauwehea x^{2}-\frac{13}{6}x+\frac{169}{144}. 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}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{25}{144}$$

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

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{25}{144}}$$

Whakarūnātia.

$$x-\frac{13}{12}=\frac{5}{12} x-\frac{13}{12}=-\frac{5}{12}$$

Me tāpiri \frac{13}{12} ki ngā taha e rua o te whārite.

$$x=\frac{3}{2} x=\frac{2}{3}$$