Whakaoti i te 2x(x-5)+3x=10(1/2-x)

$$2 x ( x - 5 ) + 3 x = 10 ( \frac { 1 } { 2 } - x )$$

Whakaoti mō x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

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

Pahekotia te -10x me 3x, ka -7x.

$$2x^{2}-7x=10\left(\frac{1}{2}-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te \frac{1}{2}-x.

$$2x^{2}-7x=10\times \frac{1}{2}-10x$$

Whakareatia te 10 ki te \frac{1}{2}, ka \frac{10}{2}.

$$2x^{2}-7x=\frac{10}{2}-10x$$

Whakawehea te 10 ki te 2, kia riro ko 5.

$$2x^{2}-7x=5-10x$$

Tangohia te 5 mai i ngā taha e rua.

$$2x^{2}-7x-5=-10x$$

Me tāpiri te 10x ki ngā taha e rua.

$$2x^{2}-7x-5+10x=0$$

Pahekotia te -7x me 10x, ka 3x.

$$2x^{2}+3x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 3 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\times 2\left(-5\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-3±\sqrt{9-8\left(-5\right)}}{2\times 2}$$

Whakareatia -8 ki te -5.

$$x=\frac{-3±\sqrt{9+40}}{2\times 2}$$

Tāpiri 9 ki te 40.

$$x=\frac{-3±\sqrt{49}}{2\times 2}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-3±7}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-3±7}{4}$$

Nā, me whakaoti te whārite x=\frac{-3±7}{4} ina he tāpiri te ±. Tāpiri -3 ki te 7.

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

Whakawehe 4 ki te 4.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-3±7}{4} ina he tango te ±. Tango 7 mai i -3.

$$x=-\frac{10}{4}$$

Whakahekea te hautanga \frac{-10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

$$x=1 x=-\frac{5}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

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

Pahekotia te -10x me 3x, ka -7x.

$$2x^{2}-7x=10\left(\frac{1}{2}-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te \frac{1}{2}-x.

$$2x^{2}-7x=10\times \frac{1}{2}-10x$$

Whakareatia te 10 ki te \frac{1}{2}, ka \frac{10}{2}.

$$2x^{2}-7x=\frac{10}{2}-10x$$

Whakawehea te 10 ki te 2, kia riro ko 5.

$$2x^{2}-7x=5-10x$$

Me tāpiri te 10x ki ngā taha e rua.

$$2x^{2}-7x+10x=5$$

Pahekotia te -7x me 10x, ka 3x.

$$2x^{2}+3x=5$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{5}{2}+\left(\frac{3}{4}\right)^{2}$$

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

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}$$

Tāpiri \frac{5}{2} ki te \frac{9}{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.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{49}{16}$$

Tauwehea te x^{2}+\frac{3}{2}x+\frac{9}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{49}{16}$$

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

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Whakarūnātia.

$$x+\frac{3}{4}=\frac{7}{4} x+\frac{3}{4}=-\frac{7}{4}$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

$$x=1 x=-\frac{5}{2}$$