Whakaoti i te 7x^2+5x-78=0

$$7 x ^ { 2 } + 5 x - 78 = 0$$

Whakaoti mō x

$$x = -\frac{26}{7} = -3\frac{5}{7} \approx -3.714285714$$

$$x=3$$

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 7x^{2}+ax+bx-78. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=7\left(-78\right)=-546$$

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 -546.

$$-1,546 -2,273 -3,182 -6,91 -7,78 -13,42 -14,39 -21,26$$

Tātaihia te tapeke mō ia takirua.

$$-1+546=545 -2+273=271 -3+182=179 -6+91=85 -7+78=71 -13+42=29 -14+39=25 -21+26=5$$

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

$$a=-21 b=26$$

Tuhia anō te 7x^{2}+5x-78 hei \left(7x^{2}-21x\right)+\left(26x-78\right).

$$\left(7x^{2}-21x\right)+\left(26x-78\right)$$

Tauwehea te 7x i te tuatahi me te 26 i te rōpū tuarua.

$$7x\left(x-3\right)+26\left(x-3\right)$$

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

$$\left(x-3\right)\left(7x+26\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te 7x+26=0.

$$x=3 x=-\frac{26}{7}$$

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.

$$7x^{2}+5x-78=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 5 mō b, me -78 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 7\left(-78\right)}}{2\times 7}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 7\left(-78\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-5±\sqrt{25-28\left(-78\right)}}{2\times 7}$$

Whakareatia -28 ki te -78.

$$x=\frac{-5±\sqrt{25+2184}}{2\times 7}$$

Tāpiri 25 ki te 2184.

$$x=\frac{-5±\sqrt{2209}}{2\times 7}$$

Tuhia te pūtakerua o te 2209.

$$x=\frac{-5±47}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-5±47}{14}$$

Nā, me whakaoti te whārite x=\frac{-5±47}{14} ina he tāpiri te ±. Tāpiri -5 ki te 47.

$$x=\frac{42}{14}$$

Whakawehe 42 ki te 14.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{-5±47}{14} ina he tango te ±. Tango 47 mai i -5.

$$x=-\frac{52}{14}$$

Whakahekea te hautanga \frac{-52}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{26}{7}$$

Kua oti te whārite te whakatau.

$$x=3 x=-\frac{26}{7}$$

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.

$$7x^{2}+5x-78=0$$

Me tāpiri 78 ki ngā taha e rua o te whārite.

$$7x^{2}+5x-78-\left(-78\right)=-\left(-78\right)$$

Mā te tango i te -78 i a ia ake anō ka toe ko te 0.

$$7x^{2}+5x=-\left(-78\right)$$

Tango -78 mai i 0.

$$7x^{2}+5x=78$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}+5x}{7}=\frac{78}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}+\frac{5}{7}x=\frac{78}{7}$$

Whakawehea te \frac{5}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{14}. Nā, tāpiria te pūrua o te \frac{5}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{7}x+\left(\frac{5}{14}\right)^{2}=\frac{78}{7}+\left(\frac{5}{14}\right)^{2}$$

Pūruatia \frac{5}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=\frac{78}{7}+\frac{25}{196}$$

Tāpiri \frac{78}{7} ki te \frac{25}{196} 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.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=\frac{2209}{196}$$

Tauwehea x^{2}+\frac{5}{7}x+\frac{25}{196}. 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}.

$$\left(x+\frac{5}{14}\right)^{2}=\frac{2209}{196}$$

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

$$\sqrt{\left(x+\frac{5}{14}\right)^{2}}=\sqrt{\frac{2209}{196}}$$

Whakarūnātia.

$$x+\frac{5}{14}=\frac{47}{14} x+\frac{5}{14}=-\frac{47}{14}$$

Me tango \frac{5}{14} mai i ngā taha e rua o te whārite.

$$x=3 x=-\frac{26}{7}$$