Whakaoti i te 75x^2+35x-10=0

$$75 x ^ { 2 } + 35 x - 10 = 0$$

Whakaoti mō x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=\frac{1}{5}=0.2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 5.

$$15x^{2}+7x-2=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 15x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=15\left(-2\right)=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-3 b=10$$

Tuhia anō te 15x^{2}+7x-2 hei \left(15x^{2}-3x\right)+\left(10x-2\right).

$$\left(15x^{2}-3x\right)+\left(10x-2\right)$$

Tauwehea te 3x i te tuatahi me te 2 i te rōpū tuarua.

$$3x\left(5x-1\right)+2\left(5x-1\right)$$

Whakatauwehea atu te kīanga pātahi 5x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-1\right)\left(3x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 5x-1=0 me te 3x+2=0.

$$x=\frac{1}{5} x=-\frac{2}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$75x^{2}+35x-10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 75 mō a, 35 mō b, me -10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\times 75\left(-10\right)}}{2\times 75}$$

Pūrua 35.

$$x=\frac{-35±\sqrt{1225-4\times 75\left(-10\right)}}{2\times 75}$$

Whakareatia -4 ki te 75.

$$x=\frac{-35±\sqrt{1225-300\left(-10\right)}}{2\times 75}$$

Whakareatia -300 ki te -10.

$$x=\frac{-35±\sqrt{1225+3000}}{2\times 75}$$

Tāpiri 1225 ki te 3000.

$$x=\frac{-35±\sqrt{4225}}{2\times 75}$$

Tuhia te pūtakerua o te 4225.

$$x=\frac{-35±65}{2\times 75}$$

Whakareatia 2 ki te 75.

$$x=\frac{-35±65}{150}$$

Nā, me whakaoti te whārite x=\frac{-35±65}{150} ina he tāpiri te ±. Tāpiri -35 ki te 65.

$$x=\frac{30}{150}$$

Whakahekea te hautanga \frac{30}{150} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 30.

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

Nā, me whakaoti te whārite x=\frac{-35±65}{150} ina he tango te ±. Tango 65 mai i -35.

$$x=-\frac{100}{150}$$

Whakahekea te hautanga \frac{-100}{150} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 50.

$$x=-\frac{2}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{5} x=-\frac{2}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$75x^{2}+35x-10=0$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$75x^{2}+35x-10-\left(-10\right)=-\left(-10\right)$$

Mā te tango i te -10 i a ia ake anō ka toe ko te 0.

$$75x^{2}+35x=-\left(-10\right)$$

Tango -10 mai i 0.

$$75x^{2}+35x=10$$

Whakawehea ngā taha e rua ki te 75.

$$\frac{75x^{2}+35x}{75}=\frac{10}{75}$$

Mā te whakawehe ki te 75 ka wetekia te whakareanga ki te 75.

$$x^{2}+\frac{35}{75}x=\frac{10}{75}$$

Whakahekea te hautanga \frac{35}{75} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{7}{15}x=\frac{10}{75}$$

Whakahekea te hautanga \frac{10}{75} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{7}{15}x=\frac{2}{15}$$

Whakawehea te \frac{7}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{30}. Nā, tāpiria te pūrua o te \frac{7}{30} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{15}x+\left(\frac{7}{30}\right)^{2}=\frac{2}{15}+\left(\frac{7}{30}\right)^{2}$$

Pūruatia \frac{7}{30} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{15}x+\frac{49}{900}=\frac{2}{15}+\frac{49}{900}$$

Tāpiri \frac{2}{15} ki te \frac{49}{900} 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.

$$x^{2}+\frac{7}{15}x+\frac{49}{900}=\frac{169}{900}$$

Tauwehea x^{2}+\frac{7}{15}x+\frac{49}{900}. 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}.

$$\left(x+\frac{7}{30}\right)^{2}=\frac{169}{900}$$

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

$$\sqrt{\left(x+\frac{7}{30}\right)^{2}}=\sqrt{\frac{169}{900}}$$

Whakarūnātia.

$$x+\frac{7}{30}=\frac{13}{30} x+\frac{7}{30}=-\frac{13}{30}$$

Me tango \frac{7}{30} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{5} x=-\frac{2}{3}$$