Whakaoti i te 4(x+3)^2-12(x+3)+9=0

$$4 ( x + 3 ) ^ { 2 } - 12 ( x + 3 ) + 9 = 0$$

Whakaoti mō x

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$4\left(x^{2}+6x+9\right)-12\left(x+3\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}+6x+9.

$$4x^{2}+24x+36-12\left(x+3\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te x+3.

$$4x^{2}+24x+36-12x-36+9=0$$

Pahekotia te 24x me -12x, ka 12x.

$$4x^{2}+12x+36-36+9=0$$

Tangohia te 36 i te 36, ka 0.

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

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 tōrunga te a+b, he tōrunga 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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$4\left(x^{2}+6x+9\right)-12\left(x+3\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}+6x+9.

$$4x^{2}+24x+36-12\left(x+3\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te x+3.

$$4x^{2}+24x+36-12x-36+9=0$$

Pahekotia te 24x me -12x, ka 12x.

$$4x^{2}+12x+36-36+9=0$$

Tangohia te 36 i te 36, ka 0.

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

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}.

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

Pūrua 12.

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

Whakareatia -4 ki te 4.

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

Whakareatia -16 ki te 9.

$$x=\frac{-12±\sqrt{144-144}}{2\times 4}$$

Tāpiri 144 ki te -144.

$$x=\frac{-12±\sqrt{0}}{2\times 4}$$

Tuhia te pūtakerua o te 0.

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

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$4\left(x^{2}+6x+9\right)-12\left(x+3\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}+6x+9.

$$4x^{2}+24x+36-12\left(x+3\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te x+3.

$$4x^{2}+24x+36-12x-36+9=0$$

Pahekotia te 24x me -12x, ka 12x.

$$4x^{2}+12x+36-36+9=0$$

Tangohia te 36 i te 36, ka 0.

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

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

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.

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

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 tango \frac{3}{2} mai i 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.

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