Whakaoti i te left(3x-4right)^2=9x-12

$${ \left(3x-4 \right) }^{ 2 } = 9x-12$$

Whakaoti mō x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

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

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-4\right)^{2}.

$$9x^{2}-24x+16=9x-12$$

Tangohia te 9x mai i ngā taha e rua.

$$9x^{2}-24x+16-9x=-12$$

Pahekotia te -24x me -9x, ka -33x.

$$9x^{2}-33x+16=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$9x^{2}-33x+16+12=0$$

Tāpirihia te 16 ki te 12, ka 28.

$$9x^{2}-33x+28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 9x^{2}+ax+bx+28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-33 ab=9\times 28=252$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 252.

$$-1,-252 -2,-126 -3,-84 -4,-63 -6,-42 -7,-36 -9,-28 -12,-21 -14,-18$$

Tātaihia te tapeke mō ia takirua.

$$-1-252=-253 -2-126=-128 -3-84=-87 -4-63=-67 -6-42=-48 -7-36=-43 -9-28=-37 -12-21=-33 -14-18=-32$$

Ko te otinga te takirua ka hoatu i te tapeke -33.

$$a=-21 b=-12$$

Tuhia anō te 9x^{2}-33x+28 hei \left(9x^{2}-21x\right)+\left(-12x+28\right).

$$\left(9x^{2}-21x\right)+\left(-12x+28\right)$$

Tauwehea te 3x i te tuatahi me te -4 i te rōpū tuarua.

$$3x\left(3x-7\right)-4\left(3x-7\right)$$

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

$$\left(3x-7\right)\left(3x-4\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-7=0 me te 3x-4=0.

$$x=\frac{7}{3} x=\frac{4}{3}$$

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-4\right)^{2}.

$$9x^{2}-24x+16=9x-12$$

Tangohia te 9x mai i ngā taha e rua.

$$9x^{2}-24x+16-9x=-12$$

Pahekotia te -24x me -9x, ka -33x.

$$9x^{2}-33x+16=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$9x^{2}-33x+16+12=0$$

Tāpirihia te 16 ki te 12, ka 28.

$$9x^{2}-33x+28=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -33 mō b, me 28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 9\times 28}}{2\times 9}$$

Pūrua -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 9\times 28}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-33\right)±\sqrt{1089-36\times 28}}{2\times 9}$$

Whakareatia -36 ki te 28.

$$x=\frac{-\left(-33\right)±\sqrt{1089-1008}}{2\times 9}$$

Tāpiri 1089 ki te -1008.

$$x=\frac{-\left(-33\right)±\sqrt{81}}{2\times 9}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-33\right)±9}{2\times 9}$$

Ko te tauaro o -33 ko 33.

$$x=\frac{33±9}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{33±9}{18}$$

Nā, me whakaoti te whārite x=\frac{33±9}{18} ina he tāpiri te ±. Tāpiri 33 ki te 9.

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

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{33±9}{18} ina he tango te ±. Tango 9 mai i 33.

$$x=\frac{24}{18}$$

Whakahekea te hautanga \frac{24}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{4}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{7}{3} x=\frac{4}{3}$$

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-4\right)^{2}.

$$9x^{2}-24x+16=9x-12$$

Tangohia te 9x mai i ngā taha e rua.

$$9x^{2}-24x+16-9x=-12$$

Pahekotia te -24x me -9x, ka -33x.

$$9x^{2}-33x+16=-12$$

Tangohia te 16 mai i ngā taha e rua.

$$9x^{2}-33x=-12-16$$

Tangohia te 16 i te -12, ka -28.

$$9x^{2}-33x=-28$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-33x}{9}=-\frac{28}{9}$$

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

$$x^{2}+\left(-\frac{33}{9}\right)x=-\frac{28}{9}$$

Whakahekea te hautanga \frac{-33}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{11}{3}x=-\frac{28}{9}$$

Whakawehea te -\frac{11}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{6}. Nā, tāpiria te pūrua o te -\frac{11}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{3}x+\left(-\frac{11}{6}\right)^{2}=-\frac{28}{9}+\left(-\frac{11}{6}\right)^{2}$$

Pūruatia -\frac{11}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=-\frac{28}{9}+\frac{121}{36}$$

Tāpiri -\frac{28}{9} ki te \frac{121}{36} 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.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=\frac{1}{4}$$

Tauwehea x^{2}-\frac{11}{3}x+\frac{121}{36}. 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}.

$$\left(x-\frac{11}{6}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(x-\frac{11}{6}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$x-\frac{11}{6}=\frac{1}{2} x-\frac{11}{6}=-\frac{1}{2}$$

Me tāpiri \frac{11}{6} ki ngā taha e rua o te whārite.

$$x=\frac{7}{3} x=\frac{4}{3}$$