Whakaoti i te 4x^2-12x+9=0

$$4 x ^ { 2 } - 12 x + 9 = 0$$

Whakaoti mō x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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 4x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=4\times 9=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-6 b=-6$$

Tuhia anō te 4x^{2}-12x+9 hei \left(4x^{2}-6x\right)+\left(-6x+9\right).

$$\left(4x^{2}-6x\right)+\left(-6x+9\right)$$

Tauwehea te 2x i te tuatahi me te -3 i te rōpū tuarua.

$$2x\left(2x-3\right)-3\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(2x-3\right)$$

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te 2x-3=0.

$$x=\frac{3}{2}$$

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.

$$4x^{2}-12x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -12 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\times 9}}{2\times 4}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\times 9}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\times 9}}{2\times 4}$$

Whakareatia -16 ki te 9.

$$x=\frac{-\left(-12\right)±\sqrt{144-144}}{2\times 4}$$

Tāpiri 144 ki te -144.

$$x=\frac{-\left(-12\right)±\sqrt{0}}{2\times 4}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-12}{2\times 4}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{12}{8}$$

Whakahekea te hautanga \frac{12}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{3}{2}$$

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.

$$4x^{2}-12x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$4x^{2}-12x+9-9=-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$4x^{2}-12x=-9$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-12x}{4}=-\frac{9}{4}$$

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

$$x^{2}+\left(-\frac{12}{4}\right)x=-\frac{9}{4}$$

Whakawehe -12 ki te 4.

$$x^{2}-3x=-\frac{9}{4}$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-\frac{9}{4}+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=\frac{-9+9}{4}$$

Tāpiri -\frac{9}{4} ki te \frac{9}{4} 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.

$$x^{2}-3x+\frac{9}{4}=0$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=0$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-\frac{3}{2}=0 x-\frac{3}{2}=0$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{3}{2} x=\frac{3}{2}$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=\frac{3}{2}$$