Whakaoti i te (2x-5)(x+1)=6x

$$( 2 x - 5 ) ( x + 1 ) = 6 x$$

Whakaoti mō x

$$x=-\frac{1}{2}=-0.5$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-5 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-5=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$2x^{2}-3x-5-6x=0$$

Pahekotia te -3x me -6x, ka -9x.

$$2x^{2}-9x-5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=2\left(-5\right)=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-10 b=1$$

Tuhia anō te 2x^{2}-9x-5 hei \left(2x^{2}-10x\right)+\left(x-5\right).

$$\left(2x^{2}-10x\right)+\left(x-5\right)$$

Whakatauwehea atu 2x i te 2x^{2}-10x.

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

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

$$\left(x-5\right)\left(2x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te 2x+1=0.

$$x=5 x=-\frac{1}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-5 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-5=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$2x^{2}-3x-5-6x=0$$

Pahekotia te -3x me -6x, ka -9x.

$$2x^{2}-9x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -9 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-5\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\left(-5\right)}}{2\times 2}$$

Whakareatia -8 ki te -5.

$$x=\frac{-\left(-9\right)±\sqrt{81+40}}{2\times 2}$$

Tāpiri 81 ki te 40.

$$x=\frac{-\left(-9\right)±\sqrt{121}}{2\times 2}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-9\right)±11}{2\times 2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±11}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{9±11}{4}$$

Nā, me whakaoti te whārite x=\frac{9±11}{4} ina he tāpiri te ±. Tāpiri 9 ki te 11.

$$x=\frac{20}{4}$$

Whakawehe 20 ki te 4.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{9±11}{4} ina he tango te ±. Tango 11 mai i 9.

$$x=-\frac{2}{4}$$

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

$$x=-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=5 x=-\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-5 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-5=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$2x^{2}-3x-5-6x=0$$

Pahekotia te -3x me -6x, ka -9x.

$$2x^{2}-9x-5=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x^{2}-9x=5$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-9x}{2}=\frac{5}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}-\frac{9}{2}x=\frac{5}{2}$$

Whakawehea te -\frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{4}. Nā, tāpiria te pūrua o te -\frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=\frac{5}{2}+\left(-\frac{9}{4}\right)^{2}$$

Pūruatia -\frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{5}{2}+\frac{81}{16}$$

Tāpiri \frac{5}{2} ki te \frac{81}{16} 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.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{121}{16}$$

Tauwehea x^{2}-\frac{9}{2}x+\frac{81}{16}. 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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{121}{16}$$

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

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Whakarūnātia.

$$x-\frac{9}{4}=\frac{11}{4} x-\frac{9}{4}=-\frac{11}{4}$$

Me tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

$$x=5 x=-\frac{1}{2}$$