Whakaoti i te 14x^2-29x-15=0

$$14 { x }^{ 2 } -29x-15 = 0$$

Whakaoti mō x

$$x=-\frac{3}{7}\approx -0.428571429$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

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 14x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-29 ab=14\left(-15\right)=-210$$

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 -210.

$$1,-210 2,-105 3,-70 5,-42 6,-35 7,-30 10,-21 14,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-210=-209 2-105=-103 3-70=-67 5-42=-37 6-35=-29 7-30=-23 10-21=-11 14-15=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -29.

$$a=-35 b=6$$

Tuhia anō te 14x^{2}-29x-15 hei \left(14x^{2}-35x\right)+\left(6x-15\right).

$$\left(14x^{2}-35x\right)+\left(6x-15\right)$$

Tauwehea te 7x i te tuatahi me te 3 i te rōpū tuarua.

$$7x\left(2x-5\right)+3\left(2x-5\right)$$

Whakatauwehea atu te kīanga pātahi 2x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-5\right)\left(7x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-5=0 me te 7x+3=0.

$$x=\frac{5}{2} x=-\frac{3}{7}$$

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.

$$14x^{2}-29x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 14 mō a, -29 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 14\left(-15\right)}}{2\times 14}$$

Pūrua -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 14\left(-15\right)}}{2\times 14}$$

Whakareatia -4 ki te 14.

$$x=\frac{-\left(-29\right)±\sqrt{841-56\left(-15\right)}}{2\times 14}$$

Whakareatia -56 ki te -15.

$$x=\frac{-\left(-29\right)±\sqrt{841+840}}{2\times 14}$$

Tāpiri 841 ki te 840.

$$x=\frac{-\left(-29\right)±\sqrt{1681}}{2\times 14}$$

Tuhia te pūtakerua o te 1681.

$$x=\frac{-\left(-29\right)±41}{2\times 14}$$

Ko te tauaro o -29 ko 29.

$$x=\frac{29±41}{2\times 14}$$

Whakareatia 2 ki te 14.

$$x=\frac{29±41}{28}$$

Nā, me whakaoti te whārite x=\frac{29±41}{28} ina he tāpiri te ±. Tāpiri 29 ki te 41.

$$x=\frac{70}{28}$$

Whakahekea te hautanga \frac{70}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x=\frac{5}{2}$$

Nā, me whakaoti te whārite x=\frac{29±41}{28} ina he tango te ±. Tango 41 mai i 29.

$$x=-\frac{12}{28}$$

Whakahekea te hautanga \frac{-12}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

$$x=\frac{5}{2} x=-\frac{3}{7}$$

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.

$$14x^{2}-29x-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$14x^{2}-29x-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$14x^{2}-29x=-\left(-15\right)$$

Tango -15 mai i 0.

$$14x^{2}-29x=15$$

Whakawehea ngā taha e rua ki te 14.

$$\frac{14x^{2}-29x}{14}=\frac{15}{14}$$

Mā te whakawehe ki te 14 ka wetekia te whakareanga ki te 14.

$$x^{2}-\frac{29}{14}x=\frac{15}{14}$$

Whakawehea te -\frac{29}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{29}{28}. Nā, tāpiria te pūrua o te -\frac{29}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{29}{14}x+\left(-\frac{29}{28}\right)^{2}=\frac{15}{14}+\left(-\frac{29}{28}\right)^{2}$$

Pūruatia -\frac{29}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{29}{14}x+\frac{841}{784}=\frac{15}{14}+\frac{841}{784}$$

Tāpiri \frac{15}{14} ki te \frac{841}{784} 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.

$$x^{2}-\frac{29}{14}x+\frac{841}{784}=\frac{1681}{784}$$

Tauwehea x^{2}-\frac{29}{14}x+\frac{841}{784}. 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}.

$$\left(x-\frac{29}{28}\right)^{2}=\frac{1681}{784}$$

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

$$\sqrt{\left(x-\frac{29}{28}\right)^{2}}=\sqrt{\frac{1681}{784}}$$

Whakarūnātia.

$$x-\frac{29}{28}=\frac{41}{28} x-\frac{29}{28}=-\frac{41}{28}$$

Me tāpiri \frac{29}{28} ki ngā taha e rua o te whārite.

$$x=\frac{5}{2} x=-\frac{3}{7}$$