Whakaoti i te (x+3)^2/5+1-(3x-1)^2/5=x(2x-3)/2

$$\frac { ( x + 3 ) ^ { 2 } } { 5 } + 1 - \frac { ( 3 x - 1 ) ^ { 2 } } { 5 } = \frac { x ( 2 x - 3 ) } { 2 }$$

Whakaoti mō x

$$x=2$$

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x+3\right)^{2}+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$2\left(x^{2}+6x+9\right)+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}+6x+9.

$$2x^{2}+12x+18+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Tāpirihia te 18 ki te 10, ka 28.

$$2x^{2}+12x+28-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3x-1\right)^{2}.

$$2x^{2}+12x+28-2\left(9x^{2}-6x+1\right)=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te 9x^{2}-6x+1.

$$2x^{2}+12x+28-18x^{2}+12x-2=5x\left(2x-3\right)$$

Pahekotia te 2x^{2} me -18x^{2}, ka -16x^{2}.

$$-16x^{2}+12x+28+12x-2=5x\left(2x-3\right)$$

Pahekotia te 12x me 12x, ka 24x.

$$-16x^{2}+24x+28-2=5x\left(2x-3\right)$$

Tangohia te 2 i te 28, ka 26.

$$-16x^{2}+24x+26=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te 2x-3.

$$-16x^{2}+24x+26=10x^{2}-15x$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$-16x^{2}+24x+26-10x^{2}=-15x$$

Pahekotia te -16x^{2} me -10x^{2}, ka -26x^{2}.

$$-26x^{2}+24x+26=-15x$$

Me tāpiri te 15x ki ngā taha e rua.

$$-26x^{2}+24x+26+15x=0$$

Pahekotia te 24x me 15x, ka 39x.

$$-26x^{2}+39x+26=0$$

Whakawehea ngā taha e rua ki te 13.

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

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

$$a+b=3 ab=-2\times 2=-4$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -4.

$$-1,4 -2,2$$

Tātaihia te tapeke mō ia takirua.

$$-1+4=3 -2+2=0$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=4 b=-1$$

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

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

Whakatauwehea atu 2x i te -2x^{2}+4x.

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

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

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x+3\right)^{2}+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$2\left(x^{2}+6x+9\right)+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}+6x+9.

$$2x^{2}+12x+18+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Tāpirihia te 18 ki te 10, ka 28.

$$2x^{2}+12x+28-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3x-1\right)^{2}.

$$2x^{2}+12x+28-2\left(9x^{2}-6x+1\right)=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te 9x^{2}-6x+1.

$$2x^{2}+12x+28-18x^{2}+12x-2=5x\left(2x-3\right)$$

Pahekotia te 2x^{2} me -18x^{2}, ka -16x^{2}.

$$-16x^{2}+12x+28+12x-2=5x\left(2x-3\right)$$

Pahekotia te 12x me 12x, ka 24x.

$$-16x^{2}+24x+28-2=5x\left(2x-3\right)$$

Tangohia te 2 i te 28, ka 26.

$$-16x^{2}+24x+26=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te 2x-3.

$$-16x^{2}+24x+26=10x^{2}-15x$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$-16x^{2}+24x+26-10x^{2}=-15x$$

Pahekotia te -16x^{2} me -10x^{2}, ka -26x^{2}.

$$-26x^{2}+24x+26=-15x$$

Me tāpiri te 15x ki ngā taha e rua.

$$-26x^{2}+24x+26+15x=0$$

Pahekotia te 24x me 15x, ka 39x.

$$-26x^{2}+39x+26=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -26 mō a, 39 mō b, me 26 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-39±\sqrt{39^{2}-4\left(-26\right)\times 26}}{2\left(-26\right)}$$

Pūrua 39.

$$x=\frac{-39±\sqrt{1521-4\left(-26\right)\times 26}}{2\left(-26\right)}$$

Whakareatia -4 ki te -26.

$$x=\frac{-39±\sqrt{1521+104\times 26}}{2\left(-26\right)}$$

Whakareatia 104 ki te 26.

$$x=\frac{-39±\sqrt{1521+2704}}{2\left(-26\right)}$$

Tāpiri 1521 ki te 2704.

$$x=\frac{-39±\sqrt{4225}}{2\left(-26\right)}$$

Tuhia te pūtakerua o te 4225.

$$x=\frac{-39±65}{2\left(-26\right)}$$

Whakareatia 2 ki te -26.

$$x=\frac{-39±65}{-52}$$

Nā, me whakaoti te whārite x=\frac{-39±65}{-52} ina he tāpiri te ±. Tāpiri -39 ki te 65.

$$x=\frac{26}{-52}$$

Whakahekea te hautanga \frac{26}{-52} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 26.

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

Nā, me whakaoti te whārite x=\frac{-39±65}{-52} ina he tango te ±. Tango 65 mai i -39.

$$x=-\frac{104}{-52}$$

Whakawehe -104 ki te -52.

$$x=2$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x+3\right)^{2}+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$2\left(x^{2}+6x+9\right)+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}+6x+9.

$$2x^{2}+12x+18+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Tāpirihia te 18 ki te 10, ka 28.

$$2x^{2}+12x+28-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3x-1\right)^{2}.

$$2x^{2}+12x+28-2\left(9x^{2}-6x+1\right)=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te 9x^{2}-6x+1.

$$2x^{2}+12x+28-18x^{2}+12x-2=5x\left(2x-3\right)$$

Pahekotia te 2x^{2} me -18x^{2}, ka -16x^{2}.

$$-16x^{2}+12x+28+12x-2=5x\left(2x-3\right)$$

Pahekotia te 12x me 12x, ka 24x.

$$-16x^{2}+24x+28-2=5x\left(2x-3\right)$$

Tangohia te 2 i te 28, ka 26.

$$-16x^{2}+24x+26=5x\left(2x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te 2x-3.

$$-16x^{2}+24x+26=10x^{2}-15x$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$-16x^{2}+24x+26-10x^{2}=-15x$$

Pahekotia te -16x^{2} me -10x^{2}, ka -26x^{2}.

$$-26x^{2}+24x+26=-15x$$

Me tāpiri te 15x ki ngā taha e rua.

$$-26x^{2}+24x+26+15x=0$$

Pahekotia te 24x me 15x, ka 39x.

$$-26x^{2}+39x+26=0$$

Tangohia te 26 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-26x^{2}+39x=-26$$

Whakawehea ngā taha e rua ki te -26.

$$\frac{-26x^{2}+39x}{-26}=-\frac{26}{-26}$$

Mā te whakawehe ki te -26 ka wetekia te whakareanga ki te -26.

$$x^{2}+\frac{39}{-26}x=-\frac{26}{-26}$$

Whakahekea te hautanga \frac{39}{-26} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 13.

$$x^{2}-\frac{3}{2}x=-\frac{26}{-26}$$

Whakawehe -26 ki te -26.

$$x^{2}-\frac{3}{2}x=1$$

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=1+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

Tāpiri 1 ki te \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

Tauwehea x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{25}{16}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{5}{4} x-\frac{3}{4}=-\frac{5}{4}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

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