Whakaoti i te x-4[3(x-2)+11]=-x/3(x+5)

$$x - 4 [ 3 ( x - 2 ) + 11 ] = - \frac { x } { 3 } ( x + 5 )$$

Whakaoti mō x

$$x=-2$$

$$x=30$$

Ngā Upane Whakamahi Tauwehe

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x-12\left(3\left(x-2\right)+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

$$3x-12\left(3x-6+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tāpirihia te -6 ki te 11, ka 5.

$$3x-12\left(3x+5\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 3x+5.

$$3x-36x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Pahekotia te 3x me -36x, ka -33x.

$$-33x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tuhia te 3\left(-\frac{x}{3}\right) hei hautanga kotahi.

$$-33x-60=\frac{-3x}{3}\left(x+5\right)$$

Me whakakore te 3 me te 3.

$$-33x-60=-x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -x ki te x+5.

$$-33x-60=-x^{2}-5x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-33x-60+x^{2}=-5x$$

Me tāpiri te 5x ki ngā taha e rua.

$$-33x-60+x^{2}+5x=0$$

Pahekotia te -33x me 5x, ka -28x.

$$-28x-60+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-28x-60=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-28x-60 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.

$$a+b=-28 ab=-60$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -28.

$$a=-30 b=2$$

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

$$\left(x-30\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-30=0 me te x+2=0.

$$x=30 x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x-12\left(3\left(x-2\right)+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

$$3x-12\left(3x-6+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tāpirihia te -6 ki te 11, ka 5.

$$3x-12\left(3x+5\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 3x+5.

$$3x-36x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Pahekotia te 3x me -36x, ka -33x.

$$-33x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tuhia te 3\left(-\frac{x}{3}\right) hei hautanga kotahi.

$$-33x-60=\frac{-3x}{3}\left(x+5\right)$$

Me whakakore te 3 me te 3.

$$-33x-60=-x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -x ki te x+5.

$$-33x-60=-x^{2}-5x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-33x-60+x^{2}=-5x$$

Me tāpiri te 5x ki ngā taha e rua.

$$-33x-60+x^{2}+5x=0$$

Pahekotia te -33x me 5x, ka -28x.

$$-28x-60+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-28x-60=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-60. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-28 ab=1\left(-60\right)=-60$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -28.

$$a=-30 b=2$$

Tuhia anō te x^{2}-28x-60 hei \left(x^{2}-30x\right)+\left(2x-60\right).

$$\left(x^{2}-30x\right)+\left(2x-60\right)$$

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

$$x\left(x-30\right)+2\left(x-30\right)$$

Whakatauwehea atu te kīanga pātahi x-30 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-30\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-30=0 me te x+2=0.

$$x=30 x=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x-12\left(3\left(x-2\right)+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

$$3x-12\left(3x-6+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tāpirihia te -6 ki te 11, ka 5.

$$3x-12\left(3x+5\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 3x+5.

$$3x-36x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Pahekotia te 3x me -36x, ka -33x.

$$-33x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tuhia te 3\left(-\frac{x}{3}\right) hei hautanga kotahi.

$$-33x-60=\frac{-3x}{3}\left(x+5\right)$$

Me whakakore te 3 me te 3.

$$-33x-60=-x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -x ki te x+5.

$$-33x-60=-x^{2}-5x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-33x-60+x^{2}=-5x$$

Me tāpiri te 5x ki ngā taha e rua.

$$-33x-60+x^{2}+5x=0$$

Pahekotia te -33x me 5x, ka -28x.

$$-28x-60+x^{2}=0$$

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.

$$x^{2}-28x-60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -28 mō b, me -60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\left(-60\right)}}{2}$$

Pūrua -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\left(-60\right)}}{2}$$

Whakareatia -4 ki te -60.

$$x=\frac{-\left(-28\right)±\sqrt{784+240}}{2}$$

Tāpiri 784 ki te 240.

$$x=\frac{-\left(-28\right)±\sqrt{1024}}{2}$$

Tuhia te pūtakerua o te 1024.

$$x=\frac{-\left(-28\right)±32}{2}$$

Ko te tauaro o -28 ko 28.

$$x=\frac{28±32}{2}$$

Nā, me whakaoti te whārite x=\frac{28±32}{2} ina he tāpiri te ±. Tāpiri 28 ki te 32.

$$x=\frac{60}{2}$$

Whakawehe 60 ki te 2.

$$x=30$$

Nā, me whakaoti te whārite x=\frac{28±32}{2} ina he tango te ±. Tango 32 mai i 28.

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

Whakawehe -4 ki te 2.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=30 x=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x-12\left(3\left(x-2\right)+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

$$3x-12\left(3x-6+11\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tāpirihia te -6 ki te 11, ka 5.

$$3x-12\left(3x+5\right)=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 3x+5.

$$3x-36x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Pahekotia te 3x me -36x, ka -33x.

$$-33x-60=3\left(-\frac{x}{3}\right)\left(x+5\right)$$

Tuhia te 3\left(-\frac{x}{3}\right) hei hautanga kotahi.

$$-33x-60=\frac{-3x}{3}\left(x+5\right)$$

Me whakakore te 3 me te 3.

$$-33x-60=-x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -x ki te x+5.

$$-33x-60=-x^{2}-5x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-33x-60+x^{2}=-5x$$

Me tāpiri te 5x ki ngā taha e rua.

$$-33x-60+x^{2}+5x=0$$

Pahekotia te -33x me 5x, ka -28x.

$$-28x-60+x^{2}=0$$

Me tāpiri te 60 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-28x+x^{2}=60$$

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.

$$x^{2}-28x=60$$

Whakawehea te -28, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -14. Nā, tāpiria te pūrua o te -14 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-28x+\left(-14\right)^{2}=60+\left(-14\right)^{2}$$

Pūrua -14.

$$x^{2}-28x+196=60+196$$

Tāpiri 60 ki te 196.

$$x^{2}-28x+196=256$$

Tauwehea x^{2}-28x+196. 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}.

$$\left(x-14\right)^{2}=256$$

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

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{256}$$

Whakarūnātia.

$$x-14=16 x-14=-16$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$x=30 x=-2$$