Whakaoti i te 56x^2-30x+1=0

$$56 { x }^{ 2 } -30x+1=0$$

Whakaoti mō x

$$x=\frac{1}{28}\approx 0.035714286$$

$$x=\frac{1}{2}=0.5$$

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 56x^{2}+ax+bx+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-30 ab=56\times 1=56$$

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 56.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

Ko te otinga te takirua ka hoatu i te tapeke -30.

$$a=-28 b=-2$$

Tuhia anō te 56x^{2}-30x+1 hei \left(56x^{2}-28x\right)+\left(-2x+1\right).

$$\left(56x^{2}-28x\right)+\left(-2x+1\right)$$

Tauwehea te 28x i te tuatahi me te -1 i te rōpū tuarua.

$$28x\left(2x-1\right)-\left(2x-1\right)$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(28x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-1=0 me te 28x-1=0.

$$x=\frac{1}{2} x=\frac{1}{28}$$

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.

$$56x^{2}-30x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 56 mō a, -30 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 56}}{2\times 56}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 56}}{2\times 56}$$

Whakareatia -4 ki te 56.

$$x=\frac{-\left(-30\right)±\sqrt{900-224}}{2\times 56}$$

Tāpiri 900 ki te -224.

$$x=\frac{-\left(-30\right)±\sqrt{676}}{2\times 56}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-\left(-30\right)±26}{2\times 56}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±26}{2\times 56}$$

Whakareatia 2 ki te 56.

$$x=\frac{30±26}{112}$$

Nā, me whakaoti te whārite x=\frac{30±26}{112} ina he tāpiri te ±. Tāpiri 30 ki te 26.

$$x=\frac{56}{112}$$

Whakahekea te hautanga \frac{56}{112} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 56.

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

Nā, me whakaoti te whārite x=\frac{30±26}{112} ina he tango te ±. Tango 26 mai i 30.

$$x=\frac{4}{112}$$

Whakahekea te hautanga \frac{4}{112} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{1}{28}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=\frac{1}{28}$$

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.

$$56x^{2}-30x+1=0$$

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

$$56x^{2}-30x+1-1=-1$$

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

$$56x^{2}-30x=-1$$

Whakawehea ngā taha e rua ki te 56.

$$\frac{56x^{2}-30x}{56}=-\frac{1}{56}$$

Mā te whakawehe ki te 56 ka wetekia te whakareanga ki te 56.

$$x^{2}+\left(-\frac{30}{56}\right)x=-\frac{1}{56}$$

Whakahekea te hautanga \frac{-30}{56} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{15}{28}x=-\frac{1}{56}$$

Whakawehea te -\frac{15}{28}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{56}. Nā, tāpiria te pūrua o te -\frac{15}{56} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{15}{28}x+\left(-\frac{15}{56}\right)^{2}=-\frac{1}{56}+\left(-\frac{15}{56}\right)^{2}$$

Pūruatia -\frac{15}{56} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{15}{28}x+\frac{225}{3136}=-\frac{1}{56}+\frac{225}{3136}$$

Tāpiri -\frac{1}{56} ki te \frac{225}{3136} 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.

$$x^{2}-\frac{15}{28}x+\frac{225}{3136}=\frac{169}{3136}$$

Tauwehea x^{2}-\frac{15}{28}x+\frac{225}{3136}. 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}.

$$\left(x-\frac{15}{56}\right)^{2}=\frac{169}{3136}$$

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

$$\sqrt{\left(x-\frac{15}{56}\right)^{2}}=\sqrt{\frac{169}{3136}}$$

Whakarūnātia.

$$x-\frac{15}{56}=\frac{13}{56} x-\frac{15}{56}=-\frac{13}{56}$$

Me tāpiri \frac{15}{56} ki ngā taha e rua o te whārite.

$$x=\frac{1}{2} x=\frac{1}{28}$$