Whakaoti i te 4x^2-7x+3=0

$$4 x ^ { 2 } - 7 x + 3 = 0$$

Whakaoti mō x

$$x=1$$

$$x=\frac{3}{4}=0.75$$

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 4x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=4\times 3=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-4 b=-3$$

Tuhia anō te 4x^{2}-7x+3 hei \left(4x^{2}-4x\right)+\left(-3x+3\right).

$$\left(4x^{2}-4x\right)+\left(-3x+3\right)$$

Tauwehea te 4x i te tuatahi me te -3 i te rōpū tuarua.

$$4x\left(x-1\right)-3\left(x-1\right)$$

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

$$\left(x-1\right)\left(4x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 4x-3=0.

$$x=1 x=\frac{3}{4}$$

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.

$$4x^{2}-7x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -7 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\times 3}}{2\times 4}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 4\times 3}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-7\right)±\sqrt{49-16\times 3}}{2\times 4}$$

Whakareatia -16 ki te 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-48}}{2\times 4}$$

Tāpiri 49 ki te -48.

$$x=\frac{-\left(-7\right)±\sqrt{1}}{2\times 4}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-\left(-7\right)±1}{2\times 4}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±1}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{7±1}{8}$$

Nā, me whakaoti te whārite x=\frac{7±1}{8} ina he tāpiri te ±. Tāpiri 7 ki te 1.

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

Whakawehe 8 ki te 8.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{7±1}{8} ina he tango te ±. Tango 1 mai i 7.

$$x=\frac{6}{8}$$

Whakahekea te hautanga \frac{6}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{3}{4}$$

Kua oti te whārite te whakatau.

$$x=1 x=\frac{3}{4}$$

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.

$$4x^{2}-7x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$4x^{2}-7x+3-3=-3$$

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

$$4x^{2}-7x=-3$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-7x}{4}=-\frac{3}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}-\frac{7}{4}x=-\frac{3}{4}$$

Whakawehea te -\frac{7}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{8}. Nā, tāpiria te pūrua o te -\frac{7}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=-\frac{3}{4}+\left(-\frac{7}{8}\right)^{2}$$

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

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=-\frac{3}{4}+\frac{49}{64}$$

Tāpiri -\frac{3}{4} ki te \frac{49}{64} 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.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{1}{64}$$

Tauwehea te x^{2}-\frac{7}{4}x+\frac{49}{64}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{1}{64}$$

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

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Whakarūnātia.

$$x-\frac{7}{8}=\frac{1}{8} x-\frac{7}{8}=-\frac{1}{8}$$

Me tāpiri \frac{7}{8} ki ngā taha e rua o te whārite.

$$x=1 x=\frac{3}{4}$$