Whakaoti i te (8x+3)^2=0

$$( 8 x + 3 ) ^ { 2 } = 0$$

Whakaoti mō x

$$x=-\frac{3}{8}=-0.375$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(8x+3\right)^{2}.

$$64x^{2}+48x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 64x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=48 ab=64\times 9=576$$

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 576.

$$1,576 2,288 3,192 4,144 6,96 8,72 9,64 12,48 16,36 18,32 24,24$$

Tātaihia te tapeke mō ia takirua.

$$1+576=577 2+288=290 3+192=195 4+144=148 6+96=102 8+72=80 9+64=73 12+48=60 16+36=52 18+32=50 24+24=48$$

Ko te otinga te takirua ka hoatu i te tapeke 48.

$$a=24 b=24$$

Tuhia anō te 64x^{2}+48x+9 hei \left(64x^{2}+24x\right)+\left(24x+9\right).

$$\left(64x^{2}+24x\right)+\left(24x+9\right)$$

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

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

Whakatauwehea atu te kīanga pātahi 8x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Tuhia anōtia hei pūrua huarua.

$$\left(8x+3\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te 8x+3=0.

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

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(8x+3\right)^{2}.

$$64x^{2}+48x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 64 mō a, 48 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 64\times 9}}{2\times 64}$$

Pūrua 48.

$$x=\frac{-48±\sqrt{2304-4\times 64\times 9}}{2\times 64}$$

Whakareatia -4 ki te 64.

$$x=\frac{-48±\sqrt{2304-256\times 9}}{2\times 64}$$

Whakareatia -256 ki te 9.

$$x=\frac{-48±\sqrt{2304-2304}}{2\times 64}$$

Tāpiri 2304 ki te -2304.

$$x=\frac{-48±\sqrt{0}}{2\times 64}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{48}{2\times 64}$$

Whakareatia 2 ki te 64.

$$x=-\frac{48}{128}$$

Whakahekea te hautanga \frac{-48}{128} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

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

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(8x+3\right)^{2}.

$$64x^{2}+48x+9=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$64x^{2}+48x=-9$$

Whakawehea ngā taha e rua ki te 64.

$$\frac{64x^{2}+48x}{64}=-\frac{9}{64}$$

Mā te whakawehe ki te 64 ka wetekia te whakareanga ki te 64.

$$x^{2}+\frac{48}{64}x=-\frac{9}{64}$$

Whakahekea te hautanga \frac{48}{64} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$x^{2}+\frac{3}{4}x=-\frac{9}{64}$$

Whakawehea te \frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{8}. Nā, tāpiria te pūrua o te \frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{9}{64}+\left(\frac{3}{8}\right)^{2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{-9+9}{64}$$

Tāpiri -\frac{9}{64} ki te \frac{9}{64} 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.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=0$$

Tauwehea x^{2}+\frac{3}{4}x+\frac{9}{64}. 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}.

$$\left(x+\frac{3}{8}\right)^{2}=0$$

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

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+\frac{3}{8}=0 x+\frac{3}{8}=0$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

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

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

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