Whakaoti i te (5x+8)^2=36

$$( 5 x + 8 ) ^ { 2 } = 36$$

Whakaoti mō x

$$x=-\frac{2}{5}=-0.4$$

$$x = -\frac{14}{5} = -2\frac{4}{5} = -2.8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(5x+8\right)^{2}.

$$25x^{2}+80x+64=36$$

Tangohia te 36 mai i ngā taha e rua.

$$25x^{2}+80x+64-36=0$$

Tangohia te 36 i te 64, ka 28.

$$25x^{2}+80x+28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 25x^{2}+ax+bx+28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=80 ab=25\times 28=700$$

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 700.

$$1,700 2,350 4,175 5,140 7,100 10,70 14,50 20,35 25,28$$

Tātaihia te tapeke mō ia takirua.

$$1+700=701 2+350=352 4+175=179 5+140=145 7+100=107 10+70=80 14+50=64 20+35=55 25+28=53$$

Ko te otinga te takirua ka hoatu i te tapeke 80.

$$a=10 b=70$$

Tuhia anō te 25x^{2}+80x+28 hei \left(25x^{2}+10x\right)+\left(70x+28\right).

$$\left(25x^{2}+10x\right)+\left(70x+28\right)$$

Tauwehea te 5x i te tuatahi me te 14 i te rōpū tuarua.

$$5x\left(5x+2\right)+14\left(5x+2\right)$$

Whakatauwehea atu te kīanga pātahi 5x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x+2\right)\left(5x+14\right)$$

Hei kimi otinga whārite, me whakaoti te 5x+2=0 me te 5x+14=0.

$$x=-\frac{2}{5} x=-\frac{14}{5}$$

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(5x+8\right)^{2}.

$$25x^{2}+80x+64=36$$

Tangohia te 36 mai i ngā taha e rua.

$$25x^{2}+80x+64-36=0$$

Tangohia te 36 i te 64, ka 28.

$$25x^{2}+80x+28=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, 80 mō b, me 28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times 25\times 28}}{2\times 25}$$

Pūrua 80.

$$x=\frac{-80±\sqrt{6400-4\times 25\times 28}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-80±\sqrt{6400-100\times 28}}{2\times 25}$$

Whakareatia -100 ki te 28.

$$x=\frac{-80±\sqrt{6400-2800}}{2\times 25}$$

Tāpiri 6400 ki te -2800.

$$x=\frac{-80±\sqrt{3600}}{2\times 25}$$

Tuhia te pūtakerua o te 3600.

$$x=\frac{-80±60}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{-80±60}{50}$$

Nā, me whakaoti te whārite x=\frac{-80±60}{50} ina he tāpiri te ±. Tāpiri -80 ki te 60.

$$x=-\frac{20}{50}$$

Whakahekea te hautanga \frac{-20}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{2}{5}$$

Nā, me whakaoti te whārite x=\frac{-80±60}{50} ina he tango te ±. Tango 60 mai i -80.

$$x=-\frac{140}{50}$$

Whakahekea te hautanga \frac{-140}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{14}{5}$$

Kua oti te whārite te whakatau.

$$x=-\frac{2}{5} x=-\frac{14}{5}$$

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(5x+8\right)^{2}.

$$25x^{2}+80x+64=36$$

Tangohia te 64 mai i ngā taha e rua.

$$25x^{2}+80x=36-64$$

Tangohia te 64 i te 36, ka -28.

$$25x^{2}+80x=-28$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}+80x}{25}=-\frac{28}{25}$$

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

$$x^{2}+\frac{80}{25}x=-\frac{28}{25}$$

Whakahekea te hautanga \frac{80}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{16}{5}x=-\frac{28}{25}$$

Whakawehea te \frac{16}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{8}{5}. Nā, tāpiria te pūrua o te \frac{8}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{16}{5}x+\left(\frac{8}{5}\right)^{2}=-\frac{28}{25}+\left(\frac{8}{5}\right)^{2}$$

Pūruatia \frac{8}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{16}{5}x+\frac{64}{25}=\frac{-28+64}{25}$$

Tāpiri -\frac{28}{25} ki te \frac{64}{25} 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.

$$x^{2}+\frac{16}{5}x+\frac{64}{25}=\frac{36}{25}$$

Tauwehea x^{2}+\frac{16}{5}x+\frac{64}{25}. 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}.

$$\left(x+\frac{8}{5}\right)^{2}=\frac{36}{25}$$

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

$$\sqrt{\left(x+\frac{8}{5}\right)^{2}}=\sqrt{\frac{36}{25}}$$

Whakarūnātia.

$$x+\frac{8}{5}=\frac{6}{5} x+\frac{8}{5}=-\frac{6}{5}$$

Me tango \frac{8}{5} mai i ngā taha e rua o te whārite.

$$x=-\frac{2}{5} x=-\frac{14}{5}$$