Whakaoti i te 2(3x+5)^2-10=22

$$2 ( 3 x + 5 ) ^ { 2 } - 10 = 22$$

Whakaoti mō x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x=-3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3x+5\right)^{2}.

$$2\left(9x^{2}+30x+25\right)-10=22$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 9x^{2}+30x+25.

$$18x^{2}+60x+50-10=22$$

Tangohia te 10 i te 50, ka 40.

$$18x^{2}+60x+40=22$$

Tangohia te 22 mai i ngā taha e rua.

$$18x^{2}+60x+40-22=0$$

Tangohia te 22 i te 40, ka 18.

$$18x^{2}+60x+18=0$$

Whakawehea ngā taha e rua ki te 6.

$$3x^{2}+10x+3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=3\times 3=9$$

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 9.

$$1,9 3,3$$

Tātaihia te tapeke mō ia takirua.

$$1+9=10 3+3=6$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=1 b=9$$

Tuhia anō te 3x^{2}+10x+3 hei \left(3x^{2}+x\right)+\left(9x+3\right).

$$\left(3x^{2}+x\right)+\left(9x+3\right)$$

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

$$x\left(3x+1\right)+3\left(3x+1\right)$$

Whakatauwehea atu te kīanga pātahi 3x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+1\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+1=0 me te x+3=0.

$$x=-\frac{1}{3} x=-3$$

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(3x+5\right)^{2}.

$$2\left(9x^{2}+30x+25\right)-10=22$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 9x^{2}+30x+25.

$$18x^{2}+60x+50-10=22$$

Tangohia te 10 i te 50, ka 40.

$$18x^{2}+60x+40=22$$

Tangohia te 22 mai i ngā taha e rua.

$$18x^{2}+60x+40-22=0$$

Tangohia te 22 i te 40, ka 18.

$$18x^{2}+60x+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 18 mō a, 60 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 18\times 18}}{2\times 18}$$

Pūrua 60.

$$x=\frac{-60±\sqrt{3600-4\times 18\times 18}}{2\times 18}$$

Whakareatia -4 ki te 18.

$$x=\frac{-60±\sqrt{3600-72\times 18}}{2\times 18}$$

Whakareatia -72 ki te 18.

$$x=\frac{-60±\sqrt{3600-1296}}{2\times 18}$$

Tāpiri 3600 ki te -1296.

$$x=\frac{-60±\sqrt{2304}}{2\times 18}$$

Tuhia te pūtakerua o te 2304.

$$x=\frac{-60±48}{2\times 18}$$

Whakareatia 2 ki te 18.

$$x=\frac{-60±48}{36}$$

Nā, me whakaoti te whārite x=\frac{-60±48}{36} ina he tāpiri te ±. Tāpiri -60 ki te 48.

$$x=-\frac{12}{36}$$

Whakahekea te hautanga \frac{-12}{36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$x=-\frac{1}{3}$$

Nā, me whakaoti te whārite x=\frac{-60±48}{36} ina he tango te ±. Tango 48 mai i -60.

$$x=-\frac{108}{36}$$

Whakawehe -108 ki te 36.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=-\frac{1}{3} x=-3$$

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(3x+5\right)^{2}.

$$2\left(9x^{2}+30x+25\right)-10=22$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 9x^{2}+30x+25.

$$18x^{2}+60x+50-10=22$$

Tangohia te 10 i te 50, ka 40.

$$18x^{2}+60x+40=22$$

Tangohia te 40 mai i ngā taha e rua.

$$18x^{2}+60x=22-40$$

Tangohia te 40 i te 22, ka -18.

$$18x^{2}+60x=-18$$

Whakawehea ngā taha e rua ki te 18.

$$\frac{18x^{2}+60x}{18}=-\frac{18}{18}$$

Mā te whakawehe ki te 18 ka wetekia te whakareanga ki te 18.

$$x^{2}+\frac{60}{18}x=-\frac{18}{18}$$

Whakahekea te hautanga \frac{60}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}+\frac{10}{3}x=-\frac{18}{18}$$

Whakawehe -18 ki te 18.

$$x^{2}+\frac{10}{3}x=-1$$

Whakawehea te \frac{10}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{3}. Nā, tāpiria te pūrua o te \frac{5}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia \frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{10}{3}x+\frac{25}{9}=-1+\frac{25}{9}$$

Tāpiri -1 ki te \frac{25}{9}.

$$x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{16}{9}$$

Tauwehea x^{2}+\frac{10}{3}x+\frac{25}{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}.

$$\left(x+\frac{5}{3}\right)^{2}=\frac{16}{9}$$

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

$$\sqrt{\left(x+\frac{5}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Whakarūnātia.

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

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{3} x=-3$$