Whakaoti i te 40x^2-14x+1=0

$$40 { x }^{ 2 } -14x+1=0$$

Whakaoti mō x

$$x=\frac{1}{10}=0.1$$

$$x=\frac{1}{4}=0.25$$

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 40x^{2}+ax+bx+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=40\times 1=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-10 b=-4$$

Tuhia anō te 40x^{2}-14x+1 hei \left(40x^{2}-10x\right)+\left(-4x+1\right).

$$\left(40x^{2}-10x\right)+\left(-4x+1\right)$$

Tauwehea te 10x i te tuatahi me te -1 i te rōpū tuarua.

$$10x\left(4x-1\right)-\left(4x-1\right)$$

Whakatauwehea atu te kīanga pātahi 4x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-1\right)\left(10x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 4x-1=0 me te 10x-1=0.

$$x=\frac{1}{4} x=\frac{1}{10}$$

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.

$$40x^{2}-14x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 40 mō a, -14 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 40}}{2\times 40}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 40}}{2\times 40}$$

Whakareatia -4 ki te 40.

$$x=\frac{-\left(-14\right)±\sqrt{196-160}}{2\times 40}$$

Tāpiri 196 ki te -160.

$$x=\frac{-\left(-14\right)±\sqrt{36}}{2\times 40}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-14\right)±6}{2\times 40}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±6}{2\times 40}$$

Whakareatia 2 ki te 40.

$$x=\frac{14±6}{80}$$

Nā, me whakaoti te whārite x=\frac{14±6}{80} ina he tāpiri te ±. Tāpiri 14 ki te 6.

$$x=\frac{20}{80}$$

Whakahekea te hautanga \frac{20}{80} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x=\frac{1}{4}$$

Nā, me whakaoti te whārite x=\frac{14±6}{80} ina he tango te ±. Tango 6 mai i 14.

$$x=\frac{8}{80}$$

Whakahekea te hautanga \frac{8}{80} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=\frac{1}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{4} x=\frac{1}{10}$$

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.

$$40x^{2}-14x+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$40x^{2}-14x+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$40x^{2}-14x=-1$$

Whakawehea ngā taha e rua ki te 40.

$$\frac{40x^{2}-14x}{40}=-\frac{1}{40}$$

Mā te whakawehe ki te 40 ka wetekia te whakareanga ki te 40.

$$x^{2}+\left(-\frac{14}{40}\right)x=-\frac{1}{40}$$

Whakahekea te hautanga \frac{-14}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{7}{20}x=-\frac{1}{40}$$

Whakawehea te -\frac{7}{20}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{40}. Nā, tāpiria te pūrua o te -\frac{7}{40} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{20}x+\left(-\frac{7}{40}\right)^{2}=-\frac{1}{40}+\left(-\frac{7}{40}\right)^{2}$$

Pūruatia -\frac{7}{40} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{20}x+\frac{49}{1600}=-\frac{1}{40}+\frac{49}{1600}$$

Tāpiri -\frac{1}{40} ki te \frac{49}{1600} 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.

$$x^{2}-\frac{7}{20}x+\frac{49}{1600}=\frac{9}{1600}$$

Tauwehea x^{2}-\frac{7}{20}x+\frac{49}{1600}. 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}.

$$\left(x-\frac{7}{40}\right)^{2}=\frac{9}{1600}$$

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

$$\sqrt{\left(x-\frac{7}{40}\right)^{2}}=\sqrt{\frac{9}{1600}}$$

Whakarūnātia.

$$x-\frac{7}{40}=\frac{3}{40} x-\frac{7}{40}=-\frac{3}{40}$$

Me tāpiri \frac{7}{40} ki ngā taha e rua o te whārite.

$$x=\frac{1}{4} x=\frac{1}{10}$$