Whakaoti i te 9x^2+14x+8=2x+4

$$9 x ^ { 2 } + 14 x + 8 = 2 x + 4$$

Whakaoti mō x

$$x=-\frac{2}{3}\approx -0.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 2x mai i ngā taha e rua.

$$9x^{2}+14x+8-2x=4$$

Pahekotia te 14x me -2x, ka 12x.

$$9x^{2}+12x+8=4$$

Tangohia te 4 mai i ngā taha e rua.

$$9x^{2}+12x+8-4=0$$

Tangohia te 4 i te 8, ka 4.

$$9x^{2}+12x+4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 9x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=9\times 4=36$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 36.

$$1,36 2,18 3,12 4,9 6,6$$

Tātaihia te tapeke mō ia takirua.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=6 b=6$$

Tuhia anō te 9x^{2}+12x+4 hei \left(9x^{2}+6x\right)+\left(6x+4\right).

$$\left(9x^{2}+6x\right)+\left(6x+4\right)$$

Tauwehea te 3x i te tuatahi me te 2 i te rōpū tuarua.

$$3x\left(3x+2\right)+2\left(3x+2\right)$$

Whakatauwehea atu te kīanga pātahi 3x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+2\right)\left(3x+2\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(3x+2\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te 3x+2=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 2x mai i ngā taha e rua.

$$9x^{2}+14x+8-2x=4$$

Pahekotia te 14x me -2x, ka 12x.

$$9x^{2}+12x+8=4$$

Tangohia te 4 mai i ngā taha e rua.

$$9x^{2}+12x+8-4=0$$

Tangohia te 4 i te 8, ka 4.

$$9x^{2}+12x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 12 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 9\times 4}}{2\times 9}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\times 9\times 4}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-12±\sqrt{144-36\times 4}}{2\times 9}$$

Whakareatia -36 ki te 4.

$$x=\frac{-12±\sqrt{144-144}}{2\times 9}$$

Tāpiri 144 ki te -144.

$$x=\frac{-12±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{12}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=-\frac{12}{18}$$

Whakahekea te hautanga \frac{-12}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 2x mai i ngā taha e rua.

$$9x^{2}+14x+8-2x=4$$

Pahekotia te 14x me -2x, ka 12x.

$$9x^{2}+12x+8=4$$

Tangohia te 8 mai i ngā taha e rua.

$$9x^{2}+12x=4-8$$

Tangohia te 8 i te 4, ka -4.

$$9x^{2}+12x=-4$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+12x}{9}=-\frac{4}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\frac{12}{9}x=-\frac{4}{9}$$

Whakahekea te hautanga \frac{12}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

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

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

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=-\frac{4}{9}+\left(\frac{2}{3}\right)^{2}$$

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

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

Tāpiri -\frac{4}{9} ki te \frac{4}{9} 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.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=0$$

Tauwehea x^{2}+\frac{4}{3}x+\frac{4}{9}. 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}.

$$\left(x+\frac{2}{3}\right)^{2}=0$$

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

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+\frac{2}{3}=0 x+\frac{2}{3}=0$$

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

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

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

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