Whakaoti i te (5x-4)^2=81

$$( 5 x - 4 ) ^ { 2 } = 81$$

Whakaoti mō x

$$x = \frac{13}{5} = 2\frac{3}{5} = 2.6$$

$$x=-1$$

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-4\right)^{2}.

$$25x^{2}-40x+16=81$$

Tangohia te 81 mai i ngā taha e rua.

$$25x^{2}-40x+16-81=0$$

Tangohia te 81 i te 16, ka -65.

$$25x^{2}-40x-65=0$$

Whakawehea ngā taha e rua ki te 5.

$$5x^{2}-8x-13=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx-13. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=5\left(-13\right)=-65$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -65.

$$1,-65 5,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-65=-64 5-13=-8$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-13 b=5$$

Tuhia anō te 5x^{2}-8x-13 hei \left(5x^{2}-13x\right)+\left(5x-13\right).

$$\left(5x^{2}-13x\right)+\left(5x-13\right)$$

Whakatauwehea atu x i te 5x^{2}-13x.

$$x\left(5x-13\right)+5x-13$$

Whakatauwehea atu te kīanga pātahi 5x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-13\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 5x-13=0 me te x+1=0.

$$x=\frac{13}{5} x=-1$$

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-4\right)^{2}.

$$25x^{2}-40x+16=81$$

Tangohia te 81 mai i ngā taha e rua.

$$25x^{2}-40x+16-81=0$$

Tangohia te 81 i te 16, ka -65.

$$25x^{2}-40x-65=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, -40 mō b, me -65 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 25\left(-65\right)}}{2\times 25}$$

Pūrua -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 25\left(-65\right)}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-\left(-40\right)±\sqrt{1600-100\left(-65\right)}}{2\times 25}$$

Whakareatia -100 ki te -65.

$$x=\frac{-\left(-40\right)±\sqrt{1600+6500}}{2\times 25}$$

Tāpiri 1600 ki te 6500.

$$x=\frac{-\left(-40\right)±\sqrt{8100}}{2\times 25}$$

Tuhia te pūtakerua o te 8100.

$$x=\frac{-\left(-40\right)±90}{2\times 25}$$

Ko te tauaro o -40 ko 40.

$$x=\frac{40±90}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{40±90}{50}$$

Nā, me whakaoti te whārite x=\frac{40±90}{50} ina he tāpiri te ±. Tāpiri 40 ki te 90.

$$x=\frac{130}{50}$$

Whakahekea te hautanga \frac{130}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{13}{5}$$

Nā, me whakaoti te whārite x=\frac{40±90}{50} ina he tango te ±. Tango 90 mai i 40.

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

Whakawehe -50 ki te 50.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{13}{5} x=-1$$

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-4\right)^{2}.

$$25x^{2}-40x+16=81$$

Tangohia te 16 mai i ngā taha e rua.

$$25x^{2}-40x=81-16$$

Tangohia te 16 i te 81, ka 65.

$$25x^{2}-40x=65$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}-40x}{25}=\frac{65}{25}$$

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

$$x^{2}+\left(-\frac{40}{25}\right)x=\frac{65}{25}$$

Whakahekea te hautanga \frac{-40}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{8}{5}x=\frac{65}{25}$$

Whakahekea te hautanga \frac{65}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{8}{5}x=\frac{13}{5}$$

Whakawehea te -\frac{8}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{5}. Nā, tāpiria te pūrua o te -\frac{4}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=\frac{13}{5}+\left(-\frac{4}{5}\right)^{2}$$

Pūruatia -\frac{4}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{13}{5}+\frac{16}{25}$$

Tāpiri \frac{13}{5} ki te \frac{16}{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.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{81}{25}$$

Tauwehea x^{2}-\frac{8}{5}x+\frac{16}{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}.

$$\left(x-\frac{4}{5}\right)^{2}=\frac{81}{25}$$

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

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{\frac{81}{25}}$$

Whakarūnātia.

$$x-\frac{4}{5}=\frac{9}{5} x-\frac{4}{5}=-\frac{9}{5}$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$x=\frac{13}{5} x=-1$$