Whakaoti i te 5x^2-7x=24

$$5 x ^ { 2 } - 7 x = 24$$

Whakaoti mō x

$$x = -\frac{8}{5} = -1\frac{3}{5} = -1.6$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 24 mai i ngā taha e rua.

$$5x^{2}-7x-24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=5\left(-24\right)=-120$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

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

$$a=-15 b=8$$

Tuhia anō te 5x^{2}-7x-24 hei \left(5x^{2}-15x\right)+\left(8x-24\right).

$$\left(5x^{2}-15x\right)+\left(8x-24\right)$$

Tauwehea te 5x i te tuatahi me te 8 i te rōpū tuarua.

$$5x\left(x-3\right)+8\left(x-3\right)$$

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

$$\left(x-3\right)\left(5x+8\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te 5x+8=0.

$$x=3 x=-\frac{8}{5}$$

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.

$$5x^{2}-7x=24$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$5x^{2}-7x-24=24-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$5x^{2}-7x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -7 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 5\left(-24\right)}}{2\times 5}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 5\left(-24\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-7\right)±\sqrt{49-20\left(-24\right)}}{2\times 5}$$

Whakareatia -20 ki te -24.

$$x=\frac{-\left(-7\right)±\sqrt{49+480}}{2\times 5}$$

Tāpiri 49 ki te 480.

$$x=\frac{-\left(-7\right)±\sqrt{529}}{2\times 5}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-\left(-7\right)±23}{2\times 5}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±23}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{7±23}{10}$$

Nā, me whakaoti te whārite x=\frac{7±23}{10} ina he tāpiri te ±. Tāpiri 7 ki te 23.

$$x=\frac{30}{10}$$

Whakawehe 30 ki te 10.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{7±23}{10} ina he tango te ±. Tango 23 mai i 7.

$$x=-\frac{16}{10}$$

Whakahekea te hautanga \frac{-16}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

$$x=3 x=-\frac{8}{5}$$

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.

$$5x^{2}-7x=24$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-7x}{5}=\frac{24}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{7}{5}x=\frac{24}{5}$$

Whakawehea te -\frac{7}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{10}. Nā, tāpiria te pūrua o te -\frac{7}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=\frac{24}{5}+\left(-\frac{7}{10}\right)^{2}$$

Pūruatia -\frac{7}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{24}{5}+\frac{49}{100}$$

Tāpiri \frac{24}{5} ki te \frac{49}{100} 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.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{529}{100}$$

Tauwehea x^{2}-\frac{7}{5}x+\frac{49}{100}. 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}.

$$\left(x-\frac{7}{10}\right)^{2}=\frac{529}{100}$$

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

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{\frac{529}{100}}$$

Whakarūnātia.

$$x-\frac{7}{10}=\frac{23}{10} x-\frac{7}{10}=-\frac{23}{10}$$

Me tāpiri \frac{7}{10} ki ngā taha e rua o te whārite.

$$x=3 x=-\frac{8}{5}$$