Whakaoti i te 0=7x^2+16x-15

$$0 = 7 x ^ { 2 } + 16 x - 15$$

Whakaoti mō x

$$x=-3$$

$$x=\frac{5}{7}\approx 0.714285714$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$7x^{2}+16x-15=0$$

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.

$$a+b=16 ab=7\left(-15\right)=-105$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua 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 -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 16.

$$a=-5 b=21$$

Tuhia anō te 7x^{2}+16x-15 hei \left(7x^{2}-5x\right)+\left(21x-15\right).

$$\left(7x^{2}-5x\right)+\left(21x-15\right)$$

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

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

Whakatauwehea atu te kīanga pātahi 7x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$7x^{2}+16x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 16 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 7\left(-15\right)}}{2\times 7}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 7\left(-15\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-16±\sqrt{256-28\left(-15\right)}}{2\times 7}$$

Whakareatia -28 ki te -15.

$$x=\frac{-16±\sqrt{256+420}}{2\times 7}$$

Tāpiri 256 ki te 420.

$$x=\frac{-16±\sqrt{676}}{2\times 7}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-16±26}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-16±26}{14}$$

Nā, me whakaoti te whārite x=\frac{-16±26}{14} ina he tāpiri te ±. Tāpiri -16 ki te 26.

$$x=\frac{10}{14}$$

Whakahekea te hautanga \frac{10}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-16±26}{14} ina he tango te ±. Tango 26 mai i -16.

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

Whakawehe -42 ki te 14.

$$x=-3$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$7x^{2}+16x-15=0$$

Me tāpiri te 15 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$7x^{2}+16x=15$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}+16x}{7}=\frac{15}{7}$$

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

$$x^{2}+\frac{16}{7}x=\frac{15}{7}$$

Whakawehea te \frac{16}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{8}{7}. Nā, tāpiria te pūrua o te \frac{8}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{16}{7}x+\left(\frac{8}{7}\right)^{2}=\frac{15}{7}+\left(\frac{8}{7}\right)^{2}$$

Pūruatia \frac{8}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=\frac{15}{7}+\frac{64}{49}$$

Tāpiri \frac{15}{7} ki te \frac{64}{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.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=\frac{169}{49}$$

Tauwehea x^{2}+\frac{16}{7}x+\frac{64}{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}.

$$\left(x+\frac{8}{7}\right)^{2}=\frac{169}{49}$$

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

$$\sqrt{\left(x+\frac{8}{7}\right)^{2}}=\sqrt{\frac{169}{49}}$$

Whakarūnātia.

$$x+\frac{8}{7}=\frac{13}{7} x+\frac{8}{7}=-\frac{13}{7}$$

Me tango \frac{8}{7} mai i ngā taha e rua o te whārite.

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