Whakaoti i te 4x^2+24x+32=0

$$4 x ^ { 2 } + 24 x + 32 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 4.

$$x^{2}+6x+8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=1\times 8=8$$

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 8.

$$1,8 2,4$$

Tātaihia te tapeke mō ia takirua.

$$1+8=9 2+4=6$$

Ko te otinga te takirua ka hoatu i te tapeke 6.

$$a=2 b=4$$

Tuhia anō te x^{2}+6x+8 hei \left(x^{2}+2x\right)+\left(4x+8\right).

$$\left(x^{2}+2x\right)+\left(4x+8\right)$$

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

$$x\left(x+2\right)+4\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x+2=0 me te x+4=0.

$$x=-2 x=-4$$

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.

$$4x^{2}+24x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 24 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 4\times 32}}{2\times 4}$$

Pūrua 24.

$$x=\frac{-24±\sqrt{576-4\times 4\times 32}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-24±\sqrt{576-16\times 32}}{2\times 4}$$

Whakareatia -16 ki te 32.

$$x=\frac{-24±\sqrt{576-512}}{2\times 4}$$

Tāpiri 576 ki te -512.

$$x=\frac{-24±\sqrt{64}}{2\times 4}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-24±8}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-24±8}{8}$$

Nā, me whakaoti te whārite x=\frac{-24±8}{8} ina he tāpiri te ±. Tāpiri -24 ki te 8.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-24±8}{8} ina he tango te ±. Tango 8 mai i -24.

$$x=-\frac{32}{8}$$

Whakawehe -32 ki te 8.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=-2 x=-4$$

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.

$$4x^{2}+24x+32=0$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$4x^{2}+24x+32-32=-32$$

Mā te tango i te 32 i a ia ake anō ka toe ko te 0.

$$4x^{2}+24x=-32$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+24x}{4}=-\frac{32}{4}$$

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

$$x^{2}+\frac{24}{4}x=-\frac{32}{4}$$

Whakawehe 24 ki te 4.

$$x^{2}+6x=-\frac{32}{4}$$

Whakawehe -32 ki te 4.

$$x^{2}+6x=-8$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+6x+3^{2}=-8+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-8+9$$

Tāpiri -8 ki te 9.

$$x^{2}+6x+9=1$$

Tauwehea x^{2}+6x+9. 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}.

$$\left(x+3\right)^{2}=1$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x+3=1 x+3=-1$$

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

$$x=-2 x=-4$$