Whakaoti i te 3x-15=2x(x-5)

$$3 x - 15 = 2 x ( x - 5 )$$

Whakaoti mō x

$$x=5$$

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x-5.

$$3x-15=2x^{2}-10x$$

Tangohia te 2x^{2} mai i ngā taha e rua.

$$3x-15-2x^{2}=-10x$$

Me tāpiri te 10x ki ngā taha e rua.

$$3x-15-2x^{2}+10x=0$$

Pahekotia te 3x me 10x, ka 13x.

$$13x-15-2x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-2x^{2}+13x-15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -2x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-2\left(-15\right)=30$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 30.

$$1,30 2,15 3,10 5,6$$

Tātaihia te tapeke mō ia takirua.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

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

$$a=10 b=3$$

Tuhia anō te -2x^{2}+13x-15 hei \left(-2x^{2}+10x\right)+\left(3x-15\right).

$$\left(-2x^{2}+10x\right)+\left(3x-15\right)$$

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

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

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

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

Hei kimi otinga whārite, me whakaoti te -x+5=0 me te 2x-3=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x-5.

$$3x-15=2x^{2}-10x$$

Tangohia te 2x^{2} mai i ngā taha e rua.

$$3x-15-2x^{2}=-10x$$

Me tāpiri te 10x ki ngā taha e rua.

$$3x-15-2x^{2}+10x=0$$

Pahekotia te 3x me 10x, ka 13x.

$$13x-15-2x^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-2x^{2}+13x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 13 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-13±\sqrt{169+8\left(-15\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -15.

$$x=\frac{-13±\sqrt{169-120}}{2\left(-2\right)}$$

Tāpiri 169 ki te -120.

$$x=\frac{-13±\sqrt{49}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-13±7}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-13±7}{-4}$$

Nā, me whakaoti te whārite x=\frac{-13±7}{-4} ina he tāpiri te ±. Tāpiri -13 ki te 7.

$$x=-\frac{6}{-4}$$

Whakahekea te hautanga \frac{-6}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-13±7}{-4} ina he tango te ±. Tango 7 mai i -13.

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

Whakawehe -20 ki te -4.

$$x=5$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x-5.

$$3x-15=2x^{2}-10x$$

Tangohia te 2x^{2} mai i ngā taha e rua.

$$3x-15-2x^{2}=-10x$$

Me tāpiri te 10x ki ngā taha e rua.

$$3x-15-2x^{2}+10x=0$$

Pahekotia te 3x me 10x, ka 13x.

$$13x-15-2x^{2}=0$$

Me tāpiri te 15 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$13x-2x^{2}=15$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-2x^{2}+13x=15$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{15}{-2}$$

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

$$x^{2}+\frac{13}{-2}x=\frac{15}{-2}$$

Whakawehe 13 ki te -2.

$$x^{2}-\frac{13}{2}x=\frac{15}{-2}$$

Whakawehe 15 ki te -2.

$$x^{2}-\frac{13}{2}x=-\frac{15}{2}$$

Whakawehea te -\frac{13}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{4}. Nā, tāpiria te pūrua o te -\frac{13}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-\frac{15}{2}+\left(-\frac{13}{4}\right)^{2}$$

Pūruatia -\frac{13}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{15}{2}+\frac{169}{16}$$

Tāpiri -\frac{15}{2} ki te \frac{169}{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.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{49}{16}$$

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

$$\left(x-\frac{13}{4}\right)^{2}=\frac{49}{16}$$

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

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Whakarūnātia.

$$x-\frac{13}{4}=\frac{7}{4} x-\frac{13}{4}=-\frac{7}{4}$$

Me tāpiri \frac{13}{4} ki ngā taha e rua o te whārite.

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