Whakaoti i te 6x^2+60x+150=0

$$6 x ^ { 2 } + 60 x + 150 = 0$$

Whakaoti mō x

$$x=-5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 6.

$$x^{2}+10x+25=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=1\times 25=25$$

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 25.

$$1,25 5,5$$

Tātaihia te tapeke mō ia takirua.

$$1+25=26 5+5=10$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=5 b=5$$

Tuhia anō te x^{2}+10x+25 hei \left(x^{2}+5x\right)+\left(5x+25\right).

$$\left(x^{2}+5x\right)+\left(5x+25\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x+5\right)+5\left(x+5\right)$$

Whakatauwehea atu te kīanga pātahi x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+5\right)\left(x+5\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x+5\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te x+5=0.

$$x=-5$$

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.

$$6x^{2}+60x+150=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 60 mō b, me 150 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 6\times 150}}{2\times 6}$$

Pūrua 60.

$$x=\frac{-60±\sqrt{3600-4\times 6\times 150}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-60±\sqrt{3600-24\times 150}}{2\times 6}$$

Whakareatia -24 ki te 150.

$$x=\frac{-60±\sqrt{3600-3600}}{2\times 6}$$

Tāpiri 3600 ki te -3600.

$$x=\frac{-60±\sqrt{0}}{2\times 6}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{60}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=-\frac{60}{12}$$

Whakawehe -60 ki te 12.

$$x=-5$$

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.

$$6x^{2}+60x+150=0$$

Me tango 150 mai i ngā taha e rua o te whārite.

$$6x^{2}+60x+150-150=-150$$

Mā te tango i te 150 i a ia ake anō ka toe ko te 0.

$$6x^{2}+60x=-150$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+60x}{6}=-\frac{150}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{60}{6}x=-\frac{150}{6}$$

Whakawehe 60 ki te 6.

$$x^{2}+10x=-\frac{150}{6}$$

Whakawehe -150 ki te 6.

$$x^{2}+10x=-25$$

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+10x+5^{2}=-25+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=-25+25$$

Tāpiri -25 ki te 25.

$$x^{2}+10x+25=0$$

Tauwehea x^{2}+10x+25. 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}.

$$\left(x+5\right)^{2}=0$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+5=0 x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x=-5 x=-5$$

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

$$x=-5$$