Whakaoti i te 6x^2+7x-5=0

$$6 { x }^{ 2 } +7x-5 = 0$$

Whakaoti mō x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

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

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 6x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=6\left(-5\right)=-30$$

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 6x^{2}+7x-5 hei \left(6x^{2}-3x\right)+\left(10x-5\right).

$$\left(6x^{2}-3x\right)+\left(10x-5\right)$$

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

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

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

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

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

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

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.

$$6x^{2}+7x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 7 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 6\left(-5\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-7±\sqrt{49-24\left(-5\right)}}{2\times 6}$$

Whakareatia -24 ki te -5.

$$x=\frac{-7±\sqrt{49+120}}{2\times 6}$$

Tāpiri 49 ki te 120.

$$x=\frac{-7±\sqrt{169}}{2\times 6}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-7±13}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-7±13}{12}$$

Nā, me whakaoti te whārite x=\frac{-7±13}{12} ina he tāpiri te ±. Tāpiri -7 ki te 13.

$$x=\frac{6}{12}$$

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

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

Nā, me whakaoti te whārite x=\frac{-7±13}{12} ina he tango te ±. Tango 13 mai i -7.

$$x=-\frac{20}{12}$$

Whakahekea te hautanga \frac{-20}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

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

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.

$$6x^{2}+7x-5=0$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$6x^{2}+7x-5-\left(-5\right)=-\left(-5\right)$$

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

$$6x^{2}+7x=-\left(-5\right)$$

Tango -5 mai i 0.

$$6x^{2}+7x=5$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+7x}{6}=\frac{5}{6}$$

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

$$x^{2}+\frac{7}{6}x=\frac{5}{6}$$

Whakawehea te \frac{7}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{12}. Nā, tāpiria te pūrua o te \frac{7}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=\frac{5}{6}+\left(\frac{7}{12}\right)^{2}$$

Pūruatia \frac{7}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{5}{6}+\frac{49}{144}$$

Tāpiri \frac{5}{6} ki te \frac{49}{144} 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.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{169}{144}$$

Tauwehea x^{2}+\frac{7}{6}x+\frac{49}{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}.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{169}{144}$$

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

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Whakarūnātia.

$$x+\frac{7}{12}=\frac{13}{12} x+\frac{7}{12}=-\frac{13}{12}$$

Me tango \frac{7}{12} mai i ngā taha e rua o te whārite.

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