Whakaoti i te 42x^2-5x-3=0

$$42 x ^ { 2 } - 5 x - 3 = 0$$

Whakaoti mō x

$$x=-\frac{3}{14}\approx -0.214285714$$

$$x=\frac{1}{3}\approx 0.333333333$$

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 42x^{2}+ax+bx-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=42\left(-3\right)=-126$$

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 -126.

$$1,-126 2,-63 3,-42 6,-21 7,-18 9,-14$$

Tātaihia te tapeke mō ia takirua.

$$1-126=-125 2-63=-61 3-42=-39 6-21=-15 7-18=-11 9-14=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-14 b=9$$

Tuhia anō te 42x^{2}-5x-3 hei \left(42x^{2}-14x\right)+\left(9x-3\right).

$$\left(42x^{2}-14x\right)+\left(9x-3\right)$$

Tauwehea te 14x i te tuatahi me te 3 i te rōpū tuarua.

$$14x\left(3x-1\right)+3\left(3x-1\right)$$

Whakatauwehea atu te kīanga pātahi 3x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-1\right)\left(14x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-1=0 me te 14x+3=0.

$$x=\frac{1}{3} x=-\frac{3}{14}$$

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.

$$42x^{2}-5x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 42 mō a, -5 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 42\left(-3\right)}}{2\times 42}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 42\left(-3\right)}}{2\times 42}$$

Whakareatia -4 ki te 42.

$$x=\frac{-\left(-5\right)±\sqrt{25-168\left(-3\right)}}{2\times 42}$$

Whakareatia -168 ki te -3.

$$x=\frac{-\left(-5\right)±\sqrt{25+504}}{2\times 42}$$

Tāpiri 25 ki te 504.

$$x=\frac{-\left(-5\right)±\sqrt{529}}{2\times 42}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-\left(-5\right)±23}{2\times 42}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±23}{2\times 42}$$

Whakareatia 2 ki te 42.

$$x=\frac{5±23}{84}$$

Nā, me whakaoti te whārite x=\frac{5±23}{84} ina he tāpiri te ±. Tāpiri 5 ki te 23.

$$x=\frac{28}{84}$$

Whakahekea te hautanga \frac{28}{84} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 28.

$$x=\frac{1}{3}$$

Nā, me whakaoti te whārite x=\frac{5±23}{84} ina he tango te ±. Tango 23 mai i 5.

$$x=-\frac{18}{84}$$

Whakahekea te hautanga \frac{-18}{84} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{3}{14}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{3} x=-\frac{3}{14}$$

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.

$$42x^{2}-5x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$42x^{2}-5x-3-\left(-3\right)=-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$42x^{2}-5x=-\left(-3\right)$$

Tango -3 mai i 0.

$$42x^{2}-5x=3$$

Whakawehea ngā taha e rua ki te 42.

$$\frac{42x^{2}-5x}{42}=\frac{3}{42}$$

Mā te whakawehe ki te 42 ka wetekia te whakareanga ki te 42.

$$x^{2}-\frac{5}{42}x=\frac{3}{42}$$

Whakahekea te hautanga \frac{3}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{5}{42}x=\frac{1}{14}$$

Whakawehea te -\frac{5}{42}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{84}. Nā, tāpiria te pūrua o te -\frac{5}{84} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{42}x+\left(-\frac{5}{84}\right)^{2}=\frac{1}{14}+\left(-\frac{5}{84}\right)^{2}$$

Pūruatia -\frac{5}{84} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{42}x+\frac{25}{7056}=\frac{1}{14}+\frac{25}{7056}$$

Tāpiri \frac{1}{14} ki te \frac{25}{7056} 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.

$$x^{2}-\frac{5}{42}x+\frac{25}{7056}=\frac{529}{7056}$$

Tauwehea x^{2}-\frac{5}{42}x+\frac{25}{7056}. 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}.

$$\left(x-\frac{5}{84}\right)^{2}=\frac{529}{7056}$$

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

$$\sqrt{\left(x-\frac{5}{84}\right)^{2}}=\sqrt{\frac{529}{7056}}$$

Whakarūnātia.

$$x-\frac{5}{84}=\frac{23}{84} x-\frac{5}{84}=-\frac{23}{84}$$

Me tāpiri \frac{5}{84} ki ngā taha e rua o te whārite.

$$x=\frac{1}{3} x=-\frac{3}{14}$$