Whakaoti i te (5x-7)^2-5(2x+1)(x-2)=-x^2-[-(3x+1)]

$$( 5 x - 7 ) ^ { 2 } - 5 ( 2 x + 1 ) ( x - 2 ) = - x ^ { 2 } - [ - ( 3 x + 1 ) ]$$

Whakaoti mō x (complex solution)

$$x=\frac{29+\sqrt{87}i}{16}\approx 1.8125+0.582961191i$$

$$x=\frac{-\sqrt{87}i+29}{16}\approx 1.8125-0.582961191i$$

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(5x-7\right)^{2}.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-\left(3x+1\right)\right)$$

Hei kimi i te tauaro o 3x+1, kimihia te tauaro o ia taurangi.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-3x-1\right)$$

Hei kimi i te tauaro o -3x-1, kimihia te tauaro o ia taurangi.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}+3x+1$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}=3x+1$$

Tangohia te 3x mai i ngā taha e rua.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}-3x=1$$

Tangohia te 1 mai i ngā taha e rua.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}-3x-1=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2x+1.

$$25x^{2}-70x+49+\left(-10x-5\right)\left(x-2\right)+x^{2}-3x-1=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -10x-5 ki te x-2 ka whakakotahi i ngā kupu rite.

$$25x^{2}-70x+49-10x^{2}+15x+10+x^{2}-3x-1=0$$

Pahekotia te 25x^{2} me -10x^{2}, ka 15x^{2}.

$$15x^{2}-70x+49+15x+10+x^{2}-3x-1=0$$

Pahekotia te -70x me 15x, ka -55x.

$$15x^{2}-55x+49+10+x^{2}-3x-1=0$$

Tāpirihia te 49 ki te 10, ka 59.

$$15x^{2}-55x+59+x^{2}-3x-1=0$$

Pahekotia te 15x^{2} me x^{2}, ka 16x^{2}.

$$16x^{2}-55x+59-3x-1=0$$

Pahekotia te -55x me -3x, ka -58x.

$$16x^{2}-58x+59-1=0$$

Tangohia te 1 i te 59, ka 58.

$$16x^{2}-58x+58=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -58 mō b, me 58 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-58\right)±\sqrt{\left(-58\right)^{2}-4\times 16\times 58}}{2\times 16}$$

Pūrua -58.

$$x=\frac{-\left(-58\right)±\sqrt{3364-4\times 16\times 58}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-58\right)±\sqrt{3364-64\times 58}}{2\times 16}$$

Whakareatia -64 ki te 58.

$$x=\frac{-\left(-58\right)±\sqrt{3364-3712}}{2\times 16}$$

Tāpiri 3364 ki te -3712.

$$x=\frac{-\left(-58\right)±\sqrt{-348}}{2\times 16}$$

Tuhia te pūtakerua o te -348.

$$x=\frac{-\left(-58\right)±2\sqrt{87}i}{2\times 16}$$

Ko te tauaro o -58 ko 58.

$$x=\frac{58±2\sqrt{87}i}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{58±2\sqrt{87}i}{32}$$

Nā, me whakaoti te whārite x=\frac{58±2\sqrt{87}i}{32} ina he tāpiri te ±. Tāpiri 58 ki te 2i\sqrt{87}.

$$x=\frac{58+2\sqrt{87}i}{32}$$

Whakawehe 58+2i\sqrt{87} ki te 32.

$$x=\frac{29+\sqrt{87}i}{16}$$

Nā, me whakaoti te whārite x=\frac{58±2\sqrt{87}i}{32} ina he tango te ±. Tango 2i\sqrt{87} mai i 58.

$$x=\frac{-2\sqrt{87}i+58}{32}$$

Whakawehe 58-2i\sqrt{87} ki te 32.

$$x=\frac{-\sqrt{87}i+29}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{29+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i+29}{16}$$

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(5x-7\right)^{2}.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-\left(3x+1\right)\right)$$

Hei kimi i te tauaro o 3x+1, kimihia te tauaro o ia taurangi.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-3x-1\right)$$

Hei kimi i te tauaro o -3x-1, kimihia te tauaro o ia taurangi.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}+3x+1$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}=3x+1$$

Tangohia te 3x mai i ngā taha e rua.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}-3x=1$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2x+1.

$$25x^{2}-70x+49+\left(-10x-5\right)\left(x-2\right)+x^{2}-3x=1$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -10x-5 ki te x-2 ka whakakotahi i ngā kupu rite.

$$25x^{2}-70x+49-10x^{2}+15x+10+x^{2}-3x=1$$

Pahekotia te 25x^{2} me -10x^{2}, ka 15x^{2}.

$$15x^{2}-70x+49+15x+10+x^{2}-3x=1$$

Pahekotia te -70x me 15x, ka -55x.

$$15x^{2}-55x+49+10+x^{2}-3x=1$$

Tāpirihia te 49 ki te 10, ka 59.

$$15x^{2}-55x+59+x^{2}-3x=1$$

Pahekotia te 15x^{2} me x^{2}, ka 16x^{2}.

$$16x^{2}-55x+59-3x=1$$

Pahekotia te -55x me -3x, ka -58x.

$$16x^{2}-58x+59=1$$

Tangohia te 59 mai i ngā taha e rua.

$$16x^{2}-58x=1-59$$

Tangohia te 59 i te 1, ka -58.

$$16x^{2}-58x=-58$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}-58x}{16}=-\frac{58}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\left(-\frac{58}{16}\right)x=-\frac{58}{16}$$

Whakahekea te hautanga \frac{-58}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{29}{8}x=-\frac{58}{16}$$

Whakahekea te hautanga \frac{-58}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{29}{8}x=-\frac{29}{8}$$

Whakawehea te -\frac{29}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{29}{16}. Nā, tāpiria te pūrua o te -\frac{29}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{29}{8}x+\left(-\frac{29}{16}\right)^{2}=-\frac{29}{8}+\left(-\frac{29}{16}\right)^{2}$$

Pūruatia -\frac{29}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{29}{8}x+\frac{841}{256}=-\frac{29}{8}+\frac{841}{256}$$

Tāpiri -\frac{29}{8} ki te \frac{841}{256} 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.

$$x^{2}-\frac{29}{8}x+\frac{841}{256}=-\frac{87}{256}$$

Tauwehea x^{2}-\frac{29}{8}x+\frac{841}{256}. 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}.

$$\left(x-\frac{29}{16}\right)^{2}=-\frac{87}{256}$$

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

$$\sqrt{\left(x-\frac{29}{16}\right)^{2}}=\sqrt{-\frac{87}{256}}$$

Whakarūnātia.

$$x-\frac{29}{16}=\frac{\sqrt{87}i}{16} x-\frac{29}{16}=-\frac{\sqrt{87}i}{16}$$

Me tāpiri \frac{29}{16} ki ngā taha e rua o te whārite.

$$x=\frac{29+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i+29}{16}$$