Whakaoti i te x^2+6x+9=144

$$x ^ { 2 } + 6 x + 9 = 144$$

Whakaoti mō x

$$x=-15$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 144 mai i ngā taha e rua.

$$x^{2}+6x+9-144=0$$

Tangohia te 144 i te 9, ka -135.

$$x^{2}+6x-135=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+6x-135 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=-135$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -135.

$$-1,135 -3,45 -5,27 -9,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+135=134 -3+45=42 -5+27=22 -9+15=6$$

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

$$a=-9 b=15$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-9\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+15=0.

$$x=9 x=-15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 144 mai i ngā taha e rua.

$$x^{2}+6x+9-144=0$$

Tangohia te 144 i te 9, ka -135.

$$x^{2}+6x-135=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-135. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=1\left(-135\right)=-135$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -135.

$$-1,135 -3,45 -5,27 -9,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+135=134 -3+45=42 -5+27=22 -9+15=6$$

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

$$a=-9 b=15$$

Tuhia anō te x^{2}+6x-135 hei \left(x^{2}-9x\right)+\left(15x-135\right).

$$\left(x^{2}-9x\right)+\left(15x-135\right)$$

Tauwehea te x i te tuatahi me te 15 i te rōpū tuarua.

$$x\left(x-9\right)+15\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+15=0.

$$x=9 x=-15$$

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.

$$x^{2}+6x+9=144$$

Me tango 144 mai i ngā taha e rua o te whārite.

$$x^{2}+6x+9-144=144-144$$

Mā te tango i te 144 i a ia ake anō ka toe ko te 0.

$$x^{2}+6x+9-144=0$$

Tango 144 mai i 9.

$$x^{2}+6x-135=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me -135 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-135\right)}}{2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-135\right)}}{2}$$

Whakareatia -4 ki te -135.

$$x=\frac{-6±\sqrt{36+540}}{2}$$

Tāpiri 36 ki te 540.

$$x=\frac{-6±\sqrt{576}}{2}$$

Tuhia te pūtakerua o te 576.

$$x=\frac{-6±24}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±24}{2} ina he tāpiri te ±. Tāpiri -6 ki te 24.

$$x=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{-6±24}{2} ina he tango te ±. Tango 24 mai i -6.

$$x=-\frac{30}{2}$$

Whakawehe -30 ki te 2.

$$x=-15$$

Kua oti te whārite te whakatau.

$$x=9 x=-15$$

Ngā Upane mō te Whakaoti i te Pūrua

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}.

$$\left(x+3\right)^{2}=144$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{144}$$

Whakarūnātia.

$$x+3=12 x+3=-12$$

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

$$x=9 x=-15$$