Whakaoti i te 6x^2+5x=6

$$6 x ^ { 2 } + 5 x = 6$$

Whakaoti mō x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 6 mai i ngā taha e rua.

$$6x^{2}+5x-6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=6\left(-6\right)=-36$$

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 -36.

$$-1,36 -2,18 -3,12 -4,9 -6,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-4 b=9$$

Tuhia anō te 6x^{2}+5x-6 hei \left(6x^{2}-4x\right)+\left(9x-6\right).

$$\left(6x^{2}-4x\right)+\left(9x-6\right)$$

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

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

Whakatauwehea atu te kīanga pātahi 3x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

$$6x^{2}+5x=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$6x^{2}+5x-6=6-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$6x^{2}+5x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 5 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 6\left(-6\right)}}{2\times 6}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 6\left(-6\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-5±\sqrt{25-24\left(-6\right)}}{2\times 6}$$

Whakareatia -24 ki te -6.

$$x=\frac{-5±\sqrt{25+144}}{2\times 6}$$

Tāpiri 25 ki te 144.

$$x=\frac{-5±\sqrt{169}}{2\times 6}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-5±13}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-5±13}{12}$$

Nā, me whakaoti te whārite x=\frac{-5±13}{12} ina he tāpiri te ±. Tāpiri -5 ki te 13.

$$x=\frac{8}{12}$$

Whakahekea te hautanga \frac{8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite x=\frac{-5±13}{12} ina he tango te ±. Tango 13 mai i -5.

$$x=-\frac{18}{12}$$

Whakahekea te hautanga \frac{-18}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Kua oti te whārite te whakatau.

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

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.

$$6x^{2}+5x=6$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+5x}{6}=\frac{6}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{5}{6}x=\frac{6}{6}$$

Whakawehe 6 ki te 6.

$$x^{2}+\frac{5}{6}x=1$$

Whakawehea te \frac{5}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{12}. Nā, tāpiria te pūrua o te \frac{5}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{6}x+\left(\frac{5}{12}\right)^{2}=1+\left(\frac{5}{12}\right)^{2}$$

Pūruatia \frac{5}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{6}x+\frac{25}{144}=1+\frac{25}{144}$$

Tāpiri 1 ki te \frac{25}{144}.

$$x^{2}+\frac{5}{6}x+\frac{25}{144}=\frac{169}{144}$$

Tauwehea x^{2}+\frac{5}{6}x+\frac{25}{144}. 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}.

$$\left(x+\frac{5}{12}\right)^{2}=\frac{169}{144}$$

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

$$\sqrt{\left(x+\frac{5}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Whakarūnātia.

$$x+\frac{5}{12}=\frac{13}{12} x+\frac{5}{12}=-\frac{13}{12}$$

Me tango \frac{5}{12} mai i ngā taha e rua o te whārite.

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