Whakaoti i te 2*(3x+4)-2*(5x+2)=3:(2x+2)+4*(4x+10)

$$2 \cdot ( 3 x + 4 ) - 2 \cdot ( 5 x + 2 ) = 3 : ( 2 x + 2 ) + 4 \cdot ( 4 x + 10 )$$

Whakaoti mō x

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5}\approx -1.108452405$$

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5}\approx -1.691547595$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2\left(x+1\right).

$$2\left(3x+4\right)\times 2\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakareatia te 2 ki te 2, ka 4.

$$4\left(3x+4\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 3x+4.

$$\left(12x+16\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 12x+16 ki te x+1 ka whakakotahi i ngā kupu rite.

$$12x^{2}+28x+16-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakareatia te -2 ki te 2, ka -4.

$$12x^{2}+28x+16-4\left(5x+2\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 5x+2.

$$12x^{2}+28x+16+\left(-20x-8\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -20x-8 ki te x+1 ka whakakotahi i ngā kupu rite.

$$12x^{2}+28x+16-20x^{2}-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Pahekotia te 12x^{2} me -20x^{2}, ka -8x^{2}.

$$-8x^{2}+28x+16-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Pahekotia te 28x me -28x, ka 0.

$$-8x^{2}+16-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Tangohia te 8 i te 16, ka 8.

$$-8x^{2}+8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakareatia te 4 ki te 2, ka 8.

$$-8x^{2}+8=3+8\left(4x+10\right)\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te 4x+10.

$$-8x^{2}+8=3+\left(32x+80\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 32x+80 ki te x+1 ka whakakotahi i ngā kupu rite.

$$-8x^{2}+8=3+32x^{2}+112x+80$$

Tāpirihia te 3 ki te 80, ka 83.

$$-8x^{2}+8=83+32x^{2}+112x$$

Tangohia te 83 mai i ngā taha e rua.

$$-8x^{2}+8-83=32x^{2}+112x$$

Tangohia te 83 i te 8, ka -75.

$$-8x^{2}-75=32x^{2}+112x$$

Tangohia te 32x^{2} mai i ngā taha e rua.

$$-8x^{2}-75-32x^{2}=112x$$

Pahekotia te -8x^{2} me -32x^{2}, ka -40x^{2}.

$$-40x^{2}-75=112x$$

Tangohia te 112x mai i ngā taha e rua.

$$-40x^{2}-75-112x=0$$

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.

$$-40x^{2}-112x-75=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -40 mō a, -112 mō b, me -75 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\left(-40\right)\left(-75\right)}}{2\left(-40\right)}$$

Pūrua -112.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\left(-40\right)\left(-75\right)}}{2\left(-40\right)}$$

Whakareatia -4 ki te -40.

$$x=\frac{-\left(-112\right)±\sqrt{12544+160\left(-75\right)}}{2\left(-40\right)}$$

Whakareatia 160 ki te -75.

$$x=\frac{-\left(-112\right)±\sqrt{12544-12000}}{2\left(-40\right)}$$

Tāpiri 12544 ki te -12000.

$$x=\frac{-\left(-112\right)±\sqrt{544}}{2\left(-40\right)}$$

Tuhia te pūtakerua o te 544.

$$x=\frac{-\left(-112\right)±4\sqrt{34}}{2\left(-40\right)}$$

Ko te tauaro o -112 ko 112.

$$x=\frac{112±4\sqrt{34}}{2\left(-40\right)}$$

Whakareatia 2 ki te -40.

$$x=\frac{112±4\sqrt{34}}{-80}$$

Nā, me whakaoti te whārite x=\frac{112±4\sqrt{34}}{-80} ina he tāpiri te ±. Tāpiri 112 ki te 4\sqrt{34}.

$$x=\frac{4\sqrt{34}+112}{-80}$$

Whakawehe 112+4\sqrt{34} ki te -80.

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Nā, me whakaoti te whārite x=\frac{112±4\sqrt{34}}{-80} ina he tango te ±. Tango 4\sqrt{34} mai i 112.

$$x=\frac{112-4\sqrt{34}}{-80}$$

Whakawehe 112-4\sqrt{34} ki te -80.

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5} x=\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2\left(x+1\right).

$$2\left(3x+4\right)\times 2\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakareatia te 2 ki te 2, ka 4.

$$4\left(3x+4\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 3x+4.

$$\left(12x+16\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 12x+16 ki te x+1 ka whakakotahi i ngā kupu rite.

$$12x^{2}+28x+16-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakareatia te -2 ki te 2, ka -4.

$$12x^{2}+28x+16-4\left(5x+2\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 5x+2.

$$12x^{2}+28x+16+\left(-20x-8\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -20x-8 ki te x+1 ka whakakotahi i ngā kupu rite.

$$12x^{2}+28x+16-20x^{2}-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Pahekotia te 12x^{2} me -20x^{2}, ka -8x^{2}.

$$-8x^{2}+28x+16-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Pahekotia te 28x me -28x, ka 0.

$$-8x^{2}+16-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Tangohia te 8 i te 16, ka 8.

$$-8x^{2}+8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Whakareatia te 4 ki te 2, ka 8.

$$-8x^{2}+8=3+8\left(4x+10\right)\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te 4x+10.

$$-8x^{2}+8=3+\left(32x+80\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 32x+80 ki te x+1 ka whakakotahi i ngā kupu rite.

$$-8x^{2}+8=3+32x^{2}+112x+80$$

Tāpirihia te 3 ki te 80, ka 83.

$$-8x^{2}+8=83+32x^{2}+112x$$

Tangohia te 32x^{2} mai i ngā taha e rua.

$$-8x^{2}+8-32x^{2}=83+112x$$

Pahekotia te -8x^{2} me -32x^{2}, ka -40x^{2}.

$$-40x^{2}+8=83+112x$$

Tangohia te 112x mai i ngā taha e rua.

$$-40x^{2}+8-112x=83$$

Tangohia te 8 mai i ngā taha e rua.

$$-40x^{2}-112x=83-8$$

Tangohia te 8 i te 83, ka 75.

$$-40x^{2}-112x=75$$

Whakawehea ngā taha e rua ki te -40.

$$\frac{-40x^{2}-112x}{-40}=\frac{75}{-40}$$

Mā te whakawehe ki te -40 ka wetekia te whakareanga ki te -40.

$$x^{2}+\left(-\frac{112}{-40}\right)x=\frac{75}{-40}$$

Whakahekea te hautanga \frac{-112}{-40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x^{2}+\frac{14}{5}x=\frac{75}{-40}$$

Whakahekea te hautanga \frac{75}{-40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{14}{5}x=-\frac{15}{8}$$

Whakawehea te \frac{14}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{5}. Nā, tāpiria te pūrua o te \frac{7}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{14}{5}x+\left(\frac{7}{5}\right)^{2}=-\frac{15}{8}+\left(\frac{7}{5}\right)^{2}$$

Pūruatia \frac{7}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=-\frac{15}{8}+\frac{49}{25}$$

Tāpiri -\frac{15}{8} ki te \frac{49}{25} 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.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{17}{200}$$

Tauwehea x^{2}+\frac{14}{5}x+\frac{49}{25}. 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}.

$$\left(x+\frac{7}{5}\right)^{2}=\frac{17}{200}$$

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

$$\sqrt{\left(x+\frac{7}{5}\right)^{2}}=\sqrt{\frac{17}{200}}$$

Whakarūnātia.

$$x+\frac{7}{5}=\frac{\sqrt{34}}{20} x+\frac{7}{5}=-\frac{\sqrt{34}}{20}$$

Me tango \frac{7}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5} x=-\frac{\sqrt{34}}{20}-\frac{7}{5}$$