Whakaoti i te (5x+7)^2=16

$$( 5 x + 7 ) ^ { 2 } = 16$$

Whakaoti mō x

$$x=-\frac{3}{5}=-0.6$$

$$x = -\frac{11}{5} = -2\frac{1}{5} = -2.2$$

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+7\right)^{2}.

$$25x^{2}+70x+49=16$$

Tangohia te 16 mai i ngā taha e rua.

$$25x^{2}+70x+49-16=0$$

Tangohia te 16 i te 49, ka 33.

$$25x^{2}+70x+33=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 25x^{2}+ax+bx+33. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=70 ab=25\times 33=825$$

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 825.

$$1,825 3,275 5,165 11,75 15,55 25,33$$

Tātaihia te tapeke mō ia takirua.

$$1+825=826 3+275=278 5+165=170 11+75=86 15+55=70 25+33=58$$

Ko te otinga te takirua ka hoatu i te tapeke 70.

$$a=15 b=55$$

Tuhia anō te 25x^{2}+70x+33 hei \left(25x^{2}+15x\right)+\left(55x+33\right).

$$\left(25x^{2}+15x\right)+\left(55x+33\right)$$

Tauwehea te 5x i te tuatahi me te 11 i te rōpū tuarua.

$$5x\left(5x+3\right)+11\left(5x+3\right)$$

Whakatauwehea atu te kīanga pātahi 5x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x+3\right)\left(5x+11\right)$$

Hei kimi otinga whārite, me whakaoti te 5x+3=0 me te 5x+11=0.

$$x=-\frac{3}{5} x=-\frac{11}{5}$$

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+7\right)^{2}.

$$25x^{2}+70x+49=16$$

Tangohia te 16 mai i ngā taha e rua.

$$25x^{2}+70x+49-16=0$$

Tangohia te 16 i te 49, ka 33.

$$25x^{2}+70x+33=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, 70 mō b, me 33 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\times 25\times 33}}{2\times 25}$$

Pūrua 70.

$$x=\frac{-70±\sqrt{4900-4\times 25\times 33}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-70±\sqrt{4900-100\times 33}}{2\times 25}$$

Whakareatia -100 ki te 33.

$$x=\frac{-70±\sqrt{4900-3300}}{2\times 25}$$

Tāpiri 4900 ki te -3300.

$$x=\frac{-70±\sqrt{1600}}{2\times 25}$$

Tuhia te pūtakerua o te 1600.

$$x=\frac{-70±40}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{-70±40}{50}$$

Nā, me whakaoti te whārite x=\frac{-70±40}{50} ina he tāpiri te ±. Tāpiri -70 ki te 40.

$$x=-\frac{30}{50}$$

Whakahekea te hautanga \frac{-30}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{3}{5}$$

Nā, me whakaoti te whārite x=\frac{-70±40}{50} ina he tango te ±. Tango 40 mai i -70.

$$x=-\frac{110}{50}$$

Whakahekea te hautanga \frac{-110}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{11}{5}$$

Kua oti te whārite te whakatau.

$$x=-\frac{3}{5} x=-\frac{11}{5}$$

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+7\right)^{2}.

$$25x^{2}+70x+49=16$$

Tangohia te 49 mai i ngā taha e rua.

$$25x^{2}+70x=16-49$$

Tangohia te 49 i te 16, ka -33.

$$25x^{2}+70x=-33$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}+70x}{25}=-\frac{33}{25}$$

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

$$x^{2}+\frac{70}{25}x=-\frac{33}{25}$$

Whakahekea te hautanga \frac{70}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{14}{5}x=-\frac{33}{25}$$

Whakawehea te \frac{14}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{5}. Nā, tāpiria te pūrua o te \frac{7}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{14}{5}x+\left(\frac{7}{5}\right)^{2}=-\frac{33}{25}+\left(\frac{7}{5}\right)^{2}$$

Pūruatia \frac{7}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{-33+49}{25}$$

Tāpiri -\frac{33}{25} ki te \frac{49}{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.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{16}{25}$$

Tauwehea x^{2}+\frac{14}{5}x+\frac{49}{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}.

$$\left(x+\frac{7}{5}\right)^{2}=\frac{16}{25}$$

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

$$\sqrt{\left(x+\frac{7}{5}\right)^{2}}=\sqrt{\frac{16}{25}}$$

Whakarūnātia.

$$x+\frac{7}{5}=\frac{4}{5} x+\frac{7}{5}=-\frac{4}{5}$$

Me tango \frac{7}{5} mai i ngā taha e rua o te whārite.

$$x=-\frac{3}{5} x=-\frac{11}{5}$$