Whakaoti i te 8x^2+10x-7=0

$$8 x ^ { 2 } + 10 x - 7 = 0$$

Whakaoti mō x

$$x = -\frac{7}{4} = -1\frac{3}{4} = -1.75$$

$$x=\frac{1}{2}=0.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 8x^{2}+ax+bx-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=8\left(-7\right)=-56$$

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 -56.

$$-1,56 -2,28 -4,14 -7,8$$

Tātaihia te tapeke mō ia takirua.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

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

$$a=-4 b=14$$

Tuhia anō te 8x^{2}+10x-7 hei \left(8x^{2}-4x\right)+\left(14x-7\right).

$$\left(8x^{2}-4x\right)+\left(14x-7\right)$$

Tauwehea te 4x i te tuatahi me te 7 i te rōpū tuarua.

$$4x\left(2x-1\right)+7\left(2x-1\right)$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(4x+7\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-1=0 me te 4x+7=0.

$$x=\frac{1}{2} x=-\frac{7}{4}$$

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.

$$8x^{2}+10x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 10 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 8\left(-7\right)}}{2\times 8}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 8\left(-7\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-10±\sqrt{100-32\left(-7\right)}}{2\times 8}$$

Whakareatia -32 ki te -7.

$$x=\frac{-10±\sqrt{100+224}}{2\times 8}$$

Tāpiri 100 ki te 224.

$$x=\frac{-10±\sqrt{324}}{2\times 8}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-10±18}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-10±18}{16}$$

Nā, me whakaoti te whārite x=\frac{-10±18}{16} ina he tāpiri te ±. Tāpiri -10 ki te 18.

$$x=\frac{8}{16}$$

Whakahekea te hautanga \frac{8}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±18}{16} ina he tango te ±. Tango 18 mai i -10.

$$x=-\frac{28}{16}$$

Whakahekea te hautanga \frac{-28}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=-\frac{7}{4}$$

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.

$$8x^{2}+10x-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$8x^{2}+10x-7-\left(-7\right)=-\left(-7\right)$$

Mā te tango i te -7 i a ia ake anō ka toe ko te 0.

$$8x^{2}+10x=-\left(-7\right)$$

Tango -7 mai i 0.

$$8x^{2}+10x=7$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+10x}{8}=\frac{7}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\frac{10}{8}x=\frac{7}{8}$$

Whakahekea te hautanga \frac{10}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{5}{4}x=\frac{7}{8}$$

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

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=\frac{7}{8}+\left(\frac{5}{8}\right)^{2}$$

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

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=\frac{7}{8}+\frac{25}{64}$$

Tāpiri \frac{7}{8} ki te \frac{25}{64} 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.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=\frac{81}{64}$$

Tauwehea te x^{2}+\frac{5}{4}x+\frac{25}{64}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{8}\right)^{2}=\frac{81}{64}$$

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

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Whakarūnātia.

$$x+\frac{5}{8}=\frac{9}{8} x+\frac{5}{8}=-\frac{9}{8}$$

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

$$x=\frac{1}{2} x=-\frac{7}{4}$$