Whakaoti i te 7x^2-32x-15=0

$$7 x ^ { 2 } - 32 x - 15 = 0$$

Whakaoti mō x

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

$$x=5$$

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 7x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-32 ab=7\left(-15\right)=-105$$

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 -105.

$$1,-105 3,-35 5,-21 7,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-105=-104 3-35=-32 5-21=-16 7-15=-8$$

Ko te otinga te takirua ka hoatu i te tapeke -32.

$$a=-35 b=3$$

Tuhia anō te 7x^{2}-32x-15 hei \left(7x^{2}-35x\right)+\left(3x-15\right).

$$\left(7x^{2}-35x\right)+\left(3x-15\right)$$

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

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

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

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

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

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

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.

$$7x^{2}-32x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -32 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 7\left(-15\right)}}{2\times 7}$$

Pūrua -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 7\left(-15\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-32\right)±\sqrt{1024-28\left(-15\right)}}{2\times 7}$$

Whakareatia -28 ki te -15.

$$x=\frac{-\left(-32\right)±\sqrt{1024+420}}{2\times 7}$$

Tāpiri 1024 ki te 420.

$$x=\frac{-\left(-32\right)±\sqrt{1444}}{2\times 7}$$

Tuhia te pūtakerua o te 1444.

$$x=\frac{-\left(-32\right)±38}{2\times 7}$$

Ko te tauaro o -32 ko 32.

$$x=\frac{32±38}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{32±38}{14}$$

Nā, me whakaoti te whārite x=\frac{32±38}{14} ina he tāpiri te ±. Tāpiri 32 ki te 38.

$$x=\frac{70}{14}$$

Whakawehe 70 ki te 14.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{32±38}{14} ina he tango te ±. Tango 38 mai i 32.

$$x=-\frac{6}{14}$$

Whakahekea te hautanga \frac{-6}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$7x^{2}-32x-15=0$$

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

$$7x^{2}-32x-15-\left(-15\right)=-\left(-15\right)$$

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

$$7x^{2}-32x=-\left(-15\right)$$

Tango -15 mai i 0.

$$7x^{2}-32x=15$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-32x}{7}=\frac{15}{7}$$

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

$$x^{2}-\frac{32}{7}x=\frac{15}{7}$$

Whakawehea te -\frac{32}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{16}{7}. Nā, tāpiria te pūrua o te -\frac{16}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{32}{7}x+\left(-\frac{16}{7}\right)^{2}=\frac{15}{7}+\left(-\frac{16}{7}\right)^{2}$$

Pūruatia -\frac{16}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{32}{7}x+\frac{256}{49}=\frac{15}{7}+\frac{256}{49}$$

Tāpiri \frac{15}{7} ki te \frac{256}{49} 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.

$$x^{2}-\frac{32}{7}x+\frac{256}{49}=\frac{361}{49}$$

Tauwehea x^{2}-\frac{32}{7}x+\frac{256}{49}. 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}.

$$\left(x-\frac{16}{7}\right)^{2}=\frac{361}{49}$$

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

$$\sqrt{\left(x-\frac{16}{7}\right)^{2}}=\sqrt{\frac{361}{49}}$$

Whakarūnātia.

$$x-\frac{16}{7}=\frac{19}{7} x-\frac{16}{7}=-\frac{19}{7}$$

Me tāpiri \frac{16}{7} ki ngā taha e rua o te whārite.

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