Whakaoti i te (x+3)^2+(3x-8)(3x+8)+1=3[x(x+3)+6]

$$( x + 3 ) ^ { 2 } + ( 3 x - 8 ) ( 3 x + 8 ) + 1 = 3 [ x ( x + 3 ) + 6 ]$$

Whakaoti mō x

$$x=-3$$

$$x = \frac{24}{7} = 3\frac{3}{7} \approx 3.428571429$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$x^{2}+6x+9+\left(3x-8\right)\left(3x+8\right)+1=3\left(x\left(x+3\right)+6\right)$$

Whakaarohia te \left(3x-8\right)\left(3x+8\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 8.

$$x^{2}+6x+9+\left(3x\right)^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Whakarohaina te \left(3x\right)^{2}.

$$x^{2}+6x+9+3^{2}x^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$x^{2}+6x+9+9x^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Pahekotia te x^{2} me 9x^{2}, ka 10x^{2}.

$$10x^{2}+6x+9-64+1=3\left(x\left(x+3\right)+6\right)$$

Tangohia te 64 i te 9, ka -55.

$$10x^{2}+6x-55+1=3\left(x\left(x+3\right)+6\right)$$

Tāpirihia te -55 ki te 1, ka -54.

$$10x^{2}+6x-54=3\left(x\left(x+3\right)+6\right)$$

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

$$10x^{2}+6x-54=3\left(x^{2}+3x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x^{2}+3x+6.

$$10x^{2}+6x-54=3x^{2}+9x+18$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$10x^{2}+6x-54-3x^{2}=9x+18$$

Pahekotia te 10x^{2} me -3x^{2}, ka 7x^{2}.

$$7x^{2}+6x-54=9x+18$$

Tangohia te 9x mai i ngā taha e rua.

$$7x^{2}+6x-54-9x=18$$

Pahekotia te 6x me -9x, ka -3x.

$$7x^{2}-3x-54=18$$

Tangohia te 18 mai i ngā taha e rua.

$$7x^{2}-3x-54-18=0$$

Tangohia te 18 i te -54, ka -72.

$$7x^{2}-3x-72=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 7x^{2}+ax+bx-72. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=7\left(-72\right)=-504$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-504=-503 2-252=-250 3-168=-165 4-126=-122 6-84=-78 7-72=-65 8-63=-55 9-56=-47 12-42=-30 14-36=-22 18-28=-10 21-24=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-24 b=21$$

Tuhia anō te 7x^{2}-3x-72 hei \left(7x^{2}-24x\right)+\left(21x-72\right).

$$\left(7x^{2}-24x\right)+\left(21x-72\right)$$

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

$$x\left(7x-24\right)+3\left(7x-24\right)$$

Whakatauwehea atu te kīanga pātahi 7x-24 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-24\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 7x-24=0 me te x+3=0.

$$x=\frac{24}{7} x=-3$$

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

$$x^{2}+6x+9+\left(3x-8\right)\left(3x+8\right)+1=3\left(x\left(x+3\right)+6\right)$$

Whakaarohia te \left(3x-8\right)\left(3x+8\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 8.

$$x^{2}+6x+9+\left(3x\right)^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Whakarohaina te \left(3x\right)^{2}.

$$x^{2}+6x+9+3^{2}x^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$x^{2}+6x+9+9x^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Pahekotia te x^{2} me 9x^{2}, ka 10x^{2}.

$$10x^{2}+6x+9-64+1=3\left(x\left(x+3\right)+6\right)$$

Tangohia te 64 i te 9, ka -55.

$$10x^{2}+6x-55+1=3\left(x\left(x+3\right)+6\right)$$

Tāpirihia te -55 ki te 1, ka -54.

$$10x^{2}+6x-54=3\left(x\left(x+3\right)+6\right)$$

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

$$10x^{2}+6x-54=3\left(x^{2}+3x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x^{2}+3x+6.

$$10x^{2}+6x-54=3x^{2}+9x+18$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$10x^{2}+6x-54-3x^{2}=9x+18$$

Pahekotia te 10x^{2} me -3x^{2}, ka 7x^{2}.

$$7x^{2}+6x-54=9x+18$$

Tangohia te 9x mai i ngā taha e rua.

$$7x^{2}+6x-54-9x=18$$

Pahekotia te 6x me -9x, ka -3x.

$$7x^{2}-3x-54=18$$

Tangohia te 18 mai i ngā taha e rua.

$$7x^{2}-3x-54-18=0$$

Tangohia te 18 i te -54, ka -72.

$$7x^{2}-3x-72=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -3 mō b, me -72 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 7\left(-72\right)}}{2\times 7}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 7\left(-72\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-3\right)±\sqrt{9-28\left(-72\right)}}{2\times 7}$$

Whakareatia -28 ki te -72.

$$x=\frac{-\left(-3\right)±\sqrt{9+2016}}{2\times 7}$$

Tāpiri 9 ki te 2016.

$$x=\frac{-\left(-3\right)±\sqrt{2025}}{2\times 7}$$

Tuhia te pūtakerua o te 2025.

$$x=\frac{-\left(-3\right)±45}{2\times 7}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±45}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{3±45}{14}$$

Nā, me whakaoti te whārite x=\frac{3±45}{14} ina he tāpiri te ±. Tāpiri 3 ki te 45.

$$x=\frac{48}{14}$$

Whakahekea te hautanga \frac{48}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{24}{7}$$

Nā, me whakaoti te whārite x=\frac{3±45}{14} ina he tango te ±. Tango 45 mai i 3.

$$x=-\frac{42}{14}$$

Whakawehe -42 ki te 14.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=\frac{24}{7} x=-3$$

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

$$x^{2}+6x+9+\left(3x-8\right)\left(3x+8\right)+1=3\left(x\left(x+3\right)+6\right)$$

Whakaarohia te \left(3x-8\right)\left(3x+8\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 8.

$$x^{2}+6x+9+\left(3x\right)^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Whakarohaina te \left(3x\right)^{2}.

$$x^{2}+6x+9+3^{2}x^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$x^{2}+6x+9+9x^{2}-64+1=3\left(x\left(x+3\right)+6\right)$$

Pahekotia te x^{2} me 9x^{2}, ka 10x^{2}.

$$10x^{2}+6x+9-64+1=3\left(x\left(x+3\right)+6\right)$$

Tangohia te 64 i te 9, ka -55.

$$10x^{2}+6x-55+1=3\left(x\left(x+3\right)+6\right)$$

Tāpirihia te -55 ki te 1, ka -54.

$$10x^{2}+6x-54=3\left(x\left(x+3\right)+6\right)$$

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

$$10x^{2}+6x-54=3\left(x^{2}+3x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x^{2}+3x+6.

$$10x^{2}+6x-54=3x^{2}+9x+18$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$10x^{2}+6x-54-3x^{2}=9x+18$$

Pahekotia te 10x^{2} me -3x^{2}, ka 7x^{2}.

$$7x^{2}+6x-54=9x+18$$

Tangohia te 9x mai i ngā taha e rua.

$$7x^{2}+6x-54-9x=18$$

Pahekotia te 6x me -9x, ka -3x.

$$7x^{2}-3x-54=18$$

Me tāpiri te 54 ki ngā taha e rua.

$$7x^{2}-3x=18+54$$

Tāpirihia te 18 ki te 54, ka 72.

$$7x^{2}-3x=72$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-3x}{7}=\frac{72}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}-\frac{3}{7}x=\frac{72}{7}$$

Whakawehea te -\frac{3}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{14}. Nā, tāpiria te pūrua o te -\frac{3}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=\frac{72}{7}+\left(-\frac{3}{14}\right)^{2}$$

Pūruatia -\frac{3}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{72}{7}+\frac{9}{196}$$

Tāpiri \frac{72}{7} ki te \frac{9}{196} 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.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{2025}{196}$$

Tauwehea x^{2}-\frac{3}{7}x+\frac{9}{196}. 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}.

$$\left(x-\frac{3}{14}\right)^{2}=\frac{2025}{196}$$

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

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{\frac{2025}{196}}$$

Whakarūnātia.

$$x-\frac{3}{14}=\frac{45}{14} x-\frac{3}{14}=-\frac{45}{14}$$

Me tāpiri \frac{3}{14} ki ngā taha e rua o te whārite.

$$x=\frac{24}{7} x=-3$$