Whakaoti i te -39+(2x-3)^2=2(-10)

$$- 39 + ( 2 x - 3 ) ^ { 2 } = 2 ( - 10 )$$

Whakaoti mō x

$$x = \frac{\sqrt{19} + 3}{2} \approx 3.679449472$$

$$x=\frac{3-\sqrt{19}}{2}\approx -0.679449472$$

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-3\right)^{2}.

$$-39+4x^{2}-12x+9=2\left(-10\right)$$

Tāpirihia te -39 ki te 9, ka -30.

$$-30+4x^{2}-12x=2\left(-10\right)$$

Whakareatia te 2 ki te -10, ka -20.

$$-30+4x^{2}-12x=-20$$

Me tāpiri te 20 ki ngā taha e rua.

$$-30+4x^{2}-12x+20=0$$

Tāpirihia te -30 ki te 20, ka -10.

$$-10+4x^{2}-12x=0$$

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.

$$4x^{2}-12x-10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -12 mō b, me -10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-10\right)}}{2\times 4}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-10\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-10\right)}}{2\times 4}$$

Whakareatia -16 ki te -10.

$$x=\frac{-\left(-12\right)±\sqrt{144+160}}{2\times 4}$$

Tāpiri 144 ki te 160.

$$x=\frac{-\left(-12\right)±\sqrt{304}}{2\times 4}$$

Tuhia te pūtakerua o te 304.

$$x=\frac{-\left(-12\right)±4\sqrt{19}}{2\times 4}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±4\sqrt{19}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{12±4\sqrt{19}}{8}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{19}}{8} ina he tāpiri te ±. Tāpiri 12 ki te 4\sqrt{19}.

$$x=\frac{4\sqrt{19}+12}{8}$$

Whakawehe 12+4\sqrt{19} ki te 8.

$$x=\frac{\sqrt{19}+3}{2}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{19}}{8} ina he tango te ±. Tango 4\sqrt{19} mai i 12.

$$x=\frac{12-4\sqrt{19}}{8}$$

Whakawehe 12-4\sqrt{19} ki te 8.

$$x=\frac{3-\sqrt{19}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{19}+3}{2} x=\frac{3-\sqrt{19}}{2}$$

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-3\right)^{2}.

$$-39+4x^{2}-12x+9=2\left(-10\right)$$

Tāpirihia te -39 ki te 9, ka -30.

$$-30+4x^{2}-12x=2\left(-10\right)$$

Whakareatia te 2 ki te -10, ka -20.

$$-30+4x^{2}-12x=-20$$

Me tāpiri te 30 ki ngā taha e rua.

$$4x^{2}-12x=-20+30$$

Tāpirihia te -20 ki te 30, ka 10.

$$4x^{2}-12x=10$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-12x}{4}=\frac{10}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\left(-\frac{12}{4}\right)x=\frac{10}{4}$$

Whakawehe -12 ki te 4.

$$x^{2}-3x=\frac{10}{4}$$

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

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

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

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=\frac{5}{2}+\frac{9}{4}$$

Tāpiri \frac{5}{2} ki te \frac{9}{4} 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.

$$x^{2}-3x+\frac{9}{4}=\frac{19}{4}$$

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

$$\left(x-\frac{3}{2}\right)^{2}=\frac{19}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{19}}{2} x-\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{19}+3}{2} x=\frac{3-\sqrt{19}}{2}$$