Whakaoti i te left(2x-5right)^2+x^2+6(2x-5)-12x+20=0

$${ \left(2x-5 \right) }^{ 2 } + { x }^{ 2 } +6(2x-5)-12x+20=0$$

Whakaoti mō x

$$x=1$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2x-5\right)^{2}.

$$4x^{2}-20x+25+x^{2}+6\left(2x-5\right)-12x+20=0$$

Pahekotia te 4x^{2} me x^{2}, ka 5x^{2}.

$$5x^{2}-20x+25+6\left(2x-5\right)-12x+20=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te 2x-5.

$$5x^{2}-20x+25+12x-30-12x+20=0$$

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

$$5x^{2}-8x+25-30-12x+20=0$$

Tangohia te 30 i te 25, ka -5.

$$5x^{2}-8x-5-12x+20=0$$

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

$$5x^{2}-20x-5+20=0$$

Tāpirihia te -5 ki te 20, ka 15.

$$5x^{2}-20x+15=0$$

Whakawehea ngā taha e rua ki te 5.

$$x^{2}-4x+3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\times 3=3$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-3 b=-1$$

Tuhia anō te x^{2}-4x+3 hei \left(x^{2}-3x\right)+\left(-x+3\right).

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

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

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

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

$$\left(x-3\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te x-1=0.

$$x=3 x=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2x-5\right)^{2}.

$$4x^{2}-20x+25+x^{2}+6\left(2x-5\right)-12x+20=0$$

Pahekotia te 4x^{2} me x^{2}, ka 5x^{2}.

$$5x^{2}-20x+25+6\left(2x-5\right)-12x+20=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te 2x-5.

$$5x^{2}-20x+25+12x-30-12x+20=0$$

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

$$5x^{2}-8x+25-30-12x+20=0$$

Tangohia te 30 i te 25, ka -5.

$$5x^{2}-8x-5-12x+20=0$$

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

$$5x^{2}-20x-5+20=0$$

Tāpirihia te -5 ki te 20, ka 15.

$$5x^{2}-20x+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -20 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 5\times 15}}{2\times 5}$$

Pūrua -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 5\times 15}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-20\right)±\sqrt{400-20\times 15}}{2\times 5}$$

Whakareatia -20 ki te 15.

$$x=\frac{-\left(-20\right)±\sqrt{400-300}}{2\times 5}$$

Tāpiri 400 ki te -300.

$$x=\frac{-\left(-20\right)±\sqrt{100}}{2\times 5}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-20\right)±10}{2\times 5}$$

Ko te tauaro o -20 ko 20.

$$x=\frac{20±10}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{20±10}{10}$$

Nā, me whakaoti te whārite x=\frac{20±10}{10} ina he tāpiri te ±. Tāpiri 20 ki te 10.

$$x=\frac{30}{10}$$

Whakawehe 30 ki te 10.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{20±10}{10} ina he tango te ±. Tango 10 mai i 20.

$$x=\frac{10}{10}$$

Whakawehe 10 ki te 10.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=3 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2x-5\right)^{2}.

$$4x^{2}-20x+25+x^{2}+6\left(2x-5\right)-12x+20=0$$

Pahekotia te 4x^{2} me x^{2}, ka 5x^{2}.

$$5x^{2}-20x+25+6\left(2x-5\right)-12x+20=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te 2x-5.

$$5x^{2}-20x+25+12x-30-12x+20=0$$

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

$$5x^{2}-8x+25-30-12x+20=0$$

Tangohia te 30 i te 25, ka -5.

$$5x^{2}-8x-5-12x+20=0$$

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

$$5x^{2}-20x-5+20=0$$

Tāpirihia te -5 ki te 20, ka 15.

$$5x^{2}-20x+15=0$$

Tangohia te 15 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x^{2}-20x=-15$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-20x}{5}=-\frac{15}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{20}{5}\right)x=-\frac{15}{5}$$

Whakawehe -20 ki te 5.

$$x^{2}-4x=-\frac{15}{5}$$

Whakawehe -15 ki te 5.

$$x^{2}-4x=-3$$

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

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

Pūrua -2.

$$x^{2}-4x+4=-3+4$$

Tāpiri -3 ki te 4.

$$x^{2}-4x+4=1$$

Tauwehea x^{2}-4x+4. 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}.

$$\left(x-2\right)^{2}=1$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-2=1 x-2=-1$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=3 x=1$$