Whakaoti i te 25x^2-10x-x^2-25=0

$$25 x ^ { 2 } - 10 x - x ^ { 2 } - 25 = 0$$

Whakaoti mō x

$$x=-\frac{5}{6}\approx -0.833333333$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Pahekotia te 25x^{2} me -x^{2}, ka 24x^{2}.

$$24x^{2}-10x-25=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 24x^{2}+ax+bx-25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=24\left(-25\right)=-600$$

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 -600.

$$1,-600 2,-300 3,-200 4,-150 5,-120 6,-100 8,-75 10,-60 12,-50 15,-40 20,-30 24,-25$$

Tātaihia te tapeke mō ia takirua.

$$1-600=-599 2-300=-298 3-200=-197 4-150=-146 5-120=-115 6-100=-94 8-75=-67 10-60=-50 12-50=-38 15-40=-25 20-30=-10 24-25=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-30 b=20$$

Tuhia anō te 24x^{2}-10x-25 hei \left(24x^{2}-30x\right)+\left(20x-25\right).

$$\left(24x^{2}-30x\right)+\left(20x-25\right)$$

Tauwehea te 6x i te tuatahi me te 5 i te rōpū tuarua.

$$6x\left(4x-5\right)+5\left(4x-5\right)$$

Whakatauwehea atu te kīanga pātahi 4x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-5\right)\left(6x+5\right)$$

Hei kimi otinga whārite, me whakaoti te 4x-5=0 me te 6x+5=0.

$$x=\frac{5}{4} x=-\frac{5}{6}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te 25x^{2} me -x^{2}, ka 24x^{2}.

$$24x^{2}-10x-25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 24 mō a, -10 mō b, me -25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 24\left(-25\right)}}{2\times 24}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 24\left(-25\right)}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$x=\frac{-\left(-10\right)±\sqrt{100-96\left(-25\right)}}{2\times 24}$$

Whakareatia -96 ki te -25.

$$x=\frac{-\left(-10\right)±\sqrt{100+2400}}{2\times 24}$$

Tāpiri 100 ki te 2400.

$$x=\frac{-\left(-10\right)±\sqrt{2500}}{2\times 24}$$

Tuhia te pūtakerua o te 2500.

$$x=\frac{-\left(-10\right)±50}{2\times 24}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±50}{2\times 24}$$

Whakareatia 2 ki te 24.

$$x=\frac{10±50}{48}$$

Nā, me whakaoti te whārite x=\frac{10±50}{48} ina he tāpiri te ±. Tāpiri 10 ki te 50.

$$x=\frac{60}{48}$$

Whakahekea te hautanga \frac{60}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

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

Nā, me whakaoti te whārite x=\frac{10±50}{48} ina he tango te ±. Tango 50 mai i 10.

$$x=-\frac{40}{48}$$

Whakahekea te hautanga \frac{-40}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Kua oti te whārite te whakatau.

$$x=\frac{5}{4} x=-\frac{5}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te 25x^{2} me -x^{2}, ka 24x^{2}.

$$24x^{2}-10x-25=0$$

Me tāpiri te 25 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$24x^{2}-10x=25$$

Whakawehea ngā taha e rua ki te 24.

$$\frac{24x^{2}-10x}{24}=\frac{25}{24}$$

Mā te whakawehe ki te 24 ka wetekia te whakareanga ki te 24.

$$x^{2}+\left(-\frac{10}{24}\right)x=\frac{25}{24}$$

Whakahekea te hautanga \frac{-10}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{5}{12}x=\frac{25}{24}$$

Whakawehea te -\frac{5}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{24}. Nā, tāpiria te pūrua o te -\frac{5}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{12}x+\left(-\frac{5}{24}\right)^{2}=\frac{25}{24}+\left(-\frac{5}{24}\right)^{2}$$

Pūruatia -\frac{5}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{12}x+\frac{25}{576}=\frac{25}{24}+\frac{25}{576}$$

Tāpiri \frac{25}{24} ki te \frac{25}{576} 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.

$$x^{2}-\frac{5}{12}x+\frac{25}{576}=\frac{625}{576}$$

Tauwehea x^{2}-\frac{5}{12}x+\frac{25}{576}. 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}.

$$\left(x-\frac{5}{24}\right)^{2}=\frac{625}{576}$$

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

$$\sqrt{\left(x-\frac{5}{24}\right)^{2}}=\sqrt{\frac{625}{576}}$$

Whakarūnātia.

$$x-\frac{5}{24}=\frac{25}{24} x-\frac{5}{24}=-\frac{25}{24}$$

Me tāpiri \frac{5}{24} ki ngā taha e rua o te whārite.

$$x=\frac{5}{4} x=-\frac{5}{6}$$