Whakaoti i te 7x^2-36x+5=0

$$7 x ^ { 2 } - 36 x + 5 = 0$$

Whakaoti mō x

$$x=\frac{1}{7}\approx 0.142857143$$

$$x=5$$

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+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-36 ab=7\times 5=35$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 35.

$$-1,-35 -5,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-35=-36 -5-7=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -36.

$$a=-35 b=-1$$

Tuhia anō te 7x^{2}-36x+5 hei \left(7x^{2}-35x\right)+\left(-x+5\right).

$$\left(7x^{2}-35x\right)+\left(-x+5\right)$$

Tauwehea te 7x i te tuatahi me te -1 i te rōpū tuarua.

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

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

$$\left(x-5\right)\left(7x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te 7x-1=0.

$$x=5 x=\frac{1}{7}$$

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.

$$7x^{2}-36x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -36 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 7\times 5}}{2\times 7}$$

Pūrua -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 7\times 5}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-36\right)±\sqrt{1296-28\times 5}}{2\times 7}$$

Whakareatia -28 ki te 5.

$$x=\frac{-\left(-36\right)±\sqrt{1296-140}}{2\times 7}$$

Tāpiri 1296 ki te -140.

$$x=\frac{-\left(-36\right)±\sqrt{1156}}{2\times 7}$$

Tuhia te pūtakerua o te 1156.

$$x=\frac{-\left(-36\right)±34}{2\times 7}$$

Ko te tauaro o -36 ko 36.

$$x=\frac{36±34}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{36±34}{14}$$

Nā, me whakaoti te whārite x=\frac{36±34}{14} ina he tāpiri te ±. Tāpiri 36 ki te 34.

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

Whakawehe 70 ki te 14.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{36±34}{14} ina he tango te ±. Tango 34 mai i 36.

$$x=\frac{2}{14}$$

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

$$x=\frac{1}{7}$$

Kua oti te whārite te whakatau.

$$x=5 x=\frac{1}{7}$$

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.

$$7x^{2}-36x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$7x^{2}-36x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$7x^{2}-36x=-5$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-36x}{7}=-\frac{5}{7}$$

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

$$x^{2}-\frac{36}{7}x=-\frac{5}{7}$$

Whakawehea te -\frac{36}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{18}{7}. Nā, tāpiria te pūrua o te -\frac{18}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{36}{7}x+\left(-\frac{18}{7}\right)^{2}=-\frac{5}{7}+\left(-\frac{18}{7}\right)^{2}$$

Pūruatia -\frac{18}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{36}{7}x+\frac{324}{49}=-\frac{5}{7}+\frac{324}{49}$$

Tāpiri -\frac{5}{7} ki te \frac{324}{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.

$$x^{2}-\frac{36}{7}x+\frac{324}{49}=\frac{289}{49}$$

Tauwehea x^{2}-\frac{36}{7}x+\frac{324}{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}.

$$\left(x-\frac{18}{7}\right)^{2}=\frac{289}{49}$$

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

$$\sqrt{\left(x-\frac{18}{7}\right)^{2}}=\sqrt{\frac{289}{49}}$$

Whakarūnātia.

$$x-\frac{18}{7}=\frac{17}{7} x-\frac{18}{7}=-\frac{17}{7}$$

Me tāpiri \frac{18}{7} ki ngā taha e rua o te whārite.

$$x=5 x=\frac{1}{7}$$