Whakaoti i te (4x+6)^2=2x+3

$$( 4 x + 6 ) ^ { 2 } = 2 x + 3$$

Whakaoti mō x

$$x = -\frac{11}{8} = -1\frac{3}{8} = -1.375$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4x+6\right)^{2}.

$$16x^{2}+48x+36=2x+3$$

Tangohia te 2x mai i ngā taha e rua.

$$16x^{2}+48x+36-2x=3$$

Pahekotia te 48x me -2x, ka 46x.

$$16x^{2}+46x+36=3$$

Tangohia te 3 mai i ngā taha e rua.

$$16x^{2}+46x+36-3=0$$

Tangohia te 3 i te 36, ka 33.

$$16x^{2}+46x+33=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 16x^{2}+ax+bx+33. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=46 ab=16\times 33=528$$

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 528.

$$1,528 2,264 3,176 4,132 6,88 8,66 11,48 12,44 16,33 22,24$$

Tātaihia te tapeke mō ia takirua.

$$1+528=529 2+264=266 3+176=179 4+132=136 6+88=94 8+66=74 11+48=59 12+44=56 16+33=49 22+24=46$$

Ko te otinga te takirua ka hoatu i te tapeke 46.

$$a=22 b=24$$

Tuhia anō te 16x^{2}+46x+33 hei \left(16x^{2}+22x\right)+\left(24x+33\right).

$$\left(16x^{2}+22x\right)+\left(24x+33\right)$$

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

$$2x\left(8x+11\right)+3\left(8x+11\right)$$

Whakatauwehea atu te kīanga pātahi 8x+11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(8x+11\right)\left(2x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 8x+11=0 me te 2x+3=0.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$

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(4x+6\right)^{2}.

$$16x^{2}+48x+36=2x+3$$

Tangohia te 2x mai i ngā taha e rua.

$$16x^{2}+48x+36-2x=3$$

Pahekotia te 48x me -2x, ka 46x.

$$16x^{2}+46x+36=3$$

Tangohia te 3 mai i ngā taha e rua.

$$16x^{2}+46x+36-3=0$$

Tangohia te 3 i te 36, ka 33.

$$16x^{2}+46x+33=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, 46 mō b, me 33 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46±\sqrt{46^{2}-4\times 16\times 33}}{2\times 16}$$

Pūrua 46.

$$x=\frac{-46±\sqrt{2116-4\times 16\times 33}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-46±\sqrt{2116-64\times 33}}{2\times 16}$$

Whakareatia -64 ki te 33.

$$x=\frac{-46±\sqrt{2116-2112}}{2\times 16}$$

Tāpiri 2116 ki te -2112.

$$x=\frac{-46±\sqrt{4}}{2\times 16}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-46±2}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{-46±2}{32}$$

Nā, me whakaoti te whārite x=\frac{-46±2}{32} ina he tāpiri te ±. Tāpiri -46 ki te 2.

$$x=-\frac{44}{32}$$

Whakahekea te hautanga \frac{-44}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{11}{8}$$

Nā, me whakaoti te whārite x=\frac{-46±2}{32} ina he tango te ±. Tango 2 mai i -46.

$$x=-\frac{48}{32}$$

Whakahekea te hautanga \frac{-48}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

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

Kua oti te whārite te whakatau.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$

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(4x+6\right)^{2}.

$$16x^{2}+48x+36=2x+3$$

Tangohia te 2x mai i ngā taha e rua.

$$16x^{2}+48x+36-2x=3$$

Pahekotia te 48x me -2x, ka 46x.

$$16x^{2}+46x+36=3$$

Tangohia te 36 mai i ngā taha e rua.

$$16x^{2}+46x=3-36$$

Tangohia te 36 i te 3, ka -33.

$$16x^{2}+46x=-33$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}+46x}{16}=-\frac{33}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\frac{46}{16}x=-\frac{33}{16}$$

Whakahekea te hautanga \frac{46}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{23}{8}x=-\frac{33}{16}$$

Whakawehea te \frac{23}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{23}{16}. Nā, tāpiria te pūrua o te \frac{23}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{23}{8}x+\left(\frac{23}{16}\right)^{2}=-\frac{33}{16}+\left(\frac{23}{16}\right)^{2}$$

Pūruatia \frac{23}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{23}{8}x+\frac{529}{256}=-\frac{33}{16}+\frac{529}{256}$$

Tāpiri -\frac{33}{16} ki te \frac{529}{256} 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.

$$x^{2}+\frac{23}{8}x+\frac{529}{256}=\frac{1}{256}$$

Tauwehea te x^{2}+\frac{23}{8}x+\frac{529}{256}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{23}{16}\right)^{2}=\frac{1}{256}$$

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

$$\sqrt{\left(x+\frac{23}{16}\right)^{2}}=\sqrt{\frac{1}{256}}$$

Whakarūnātia.

$$x+\frac{23}{16}=\frac{1}{16} x+\frac{23}{16}=-\frac{1}{16}$$

Me tango \frac{23}{16} mai i ngā taha e rua o te whārite.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$