Whakaoti i te left(3x+5right)^2+5left(3x+5right)+6=0

$${ \left(3x+5 \right) }^{ 2 } +5 \left( 3x+5 \right) +6=0$$

Whakaoti mō x

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2.666666667$$

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

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}.

$$9x^{2}+30x+25+5\left(3x+5\right)+6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 3x+5.

$$9x^{2}+30x+25+15x+25+6=0$$

Pahekotia te 30x me 15x, ka 45x.

$$9x^{2}+45x+25+25+6=0$$

Tāpirihia te 25 ki te 25, ka 50.

$$9x^{2}+45x+50+6=0$$

Tāpirihia te 50 ki te 6, ka 56.

$$9x^{2}+45x+56=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 9x^{2}+ax+bx+56. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=45 ab=9\times 56=504$$

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 504.

$$1,504 2,252 3,168 4,126 6,84 7,72 8,63 9,56 12,42 14,36 18,28 21,24$$

Tātaihia te tapeke mō ia takirua.

$$1+504=505 2+252=254 3+168=171 4+126=130 6+84=90 7+72=79 8+63=71 9+56=65 12+42=54 14+36=50 18+28=46 21+24=45$$

Ko te otinga te takirua ka hoatu i te tapeke 45.

$$a=21 b=24$$

Tuhia anō te 9x^{2}+45x+56 hei \left(9x^{2}+21x\right)+\left(24x+56\right).

$$\left(9x^{2}+21x\right)+\left(24x+56\right)$$

Tauwehea te 3x i te tuatahi me te 8 i te rōpū tuarua.

$$3x\left(3x+7\right)+8\left(3x+7\right)$$

Whakatauwehea atu te kīanga pātahi 3x+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+7\right)\left(3x+8\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+7=0 me te 3x+8=0.

$$x=-\frac{7}{3} x=-\frac{8}{3}$$

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}.

$$9x^{2}+30x+25+5\left(3x+5\right)+6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 3x+5.

$$9x^{2}+30x+25+15x+25+6=0$$

Pahekotia te 30x me 15x, ka 45x.

$$9x^{2}+45x+25+25+6=0$$

Tāpirihia te 25 ki te 25, ka 50.

$$9x^{2}+45x+50+6=0$$

Tāpirihia te 50 ki te 6, ka 56.

$$9x^{2}+45x+56=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 45 mō b, me 56 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\times 9\times 56}}{2\times 9}$$

Pūrua 45.

$$x=\frac{-45±\sqrt{2025-4\times 9\times 56}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-45±\sqrt{2025-36\times 56}}{2\times 9}$$

Whakareatia -36 ki te 56.

$$x=\frac{-45±\sqrt{2025-2016}}{2\times 9}$$

Tāpiri 2025 ki te -2016.

$$x=\frac{-45±\sqrt{9}}{2\times 9}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-45±3}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-45±3}{18}$$

Nā, me whakaoti te whārite x=\frac{-45±3}{18} ina he tāpiri te ±. Tāpiri -45 ki te 3.

$$x=-\frac{42}{18}$$

Whakahekea te hautanga \frac{-42}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Nā, me whakaoti te whārite x=\frac{-45±3}{18} ina he tango te ±. Tango 3 mai i -45.

$$x=-\frac{48}{18}$$

Whakahekea te hautanga \frac{-48}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{8}{3}$$

Kua oti te whārite te whakatau.

$$x=-\frac{7}{3} x=-\frac{8}{3}$$

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}.

$$9x^{2}+30x+25+5\left(3x+5\right)+6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 3x+5.

$$9x^{2}+30x+25+15x+25+6=0$$

Pahekotia te 30x me 15x, ka 45x.

$$9x^{2}+45x+25+25+6=0$$

Tāpirihia te 25 ki te 25, ka 50.

$$9x^{2}+45x+50+6=0$$

Tāpirihia te 50 ki te 6, ka 56.

$$9x^{2}+45x+56=0$$

Tangohia te 56 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9x^{2}+45x=-56$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+45x}{9}=-\frac{56}{9}$$

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

$$x^{2}+\frac{45}{9}x=-\frac{56}{9}$$

Whakawehe 45 ki te 9.

$$x^{2}+5x=-\frac{56}{9}$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-\frac{56}{9}+\left(\frac{5}{2}\right)^{2}$$

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+5x+\frac{25}{4}=-\frac{56}{9}+\frac{25}{4}$$

Tāpiri -\frac{56}{9} ki te \frac{25}{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.

$$x^{2}+5x+\frac{25}{4}=\frac{1}{36}$$

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

$$\left(x+\frac{5}{2}\right)^{2}=\frac{1}{36}$$

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

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Whakarūnātia.

$$x+\frac{5}{2}=\frac{1}{6} x+\frac{5}{2}=-\frac{1}{6}$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$x=-\frac{7}{3} x=-\frac{8}{3}$$