Whakaoti i te 128+128(x+1)+128(x+1)^2=608

$$128 + 128 ( x + 1 ) + 128 ( x + 1 ) ^ { 2 } = 608$$

Whakaoti mō x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=\frac{1}{2}=0.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te 128 ki te x+1.

$$128+128x+128+128\left(x+1\right)^{2}=608$$

Tāpirihia te 128 ki te 128, ka 256.

$$256+128x+128\left(x+1\right)^{2}=608$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$256+128x+128\left(x^{2}+2x+1\right)=608$$

Whakamahia te āhuatanga tohatoha hei whakarea te 128 ki te x^{2}+2x+1.

$$256+128x+128x^{2}+256x+128=608$$

Pahekotia te 128x me 256x, ka 384x.

$$256+384x+128x^{2}+128=608$$

Tāpirihia te 256 ki te 128, ka 384.

$$384+384x+128x^{2}=608$$

Tangohia te 608 mai i ngā taha e rua.

$$384+384x+128x^{2}-608=0$$

Tangohia te 608 i te 384, ka -224.

$$-224+384x+128x^{2}=0$$

Whakawehea ngā taha e rua ki te 32.

$$-7+12x+4x^{2}=0$$

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

$$4x^{2}+12x-7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=4\left(-7\right)=-28$$

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 -28.

$$-1,28 -2,14 -4,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+28=27 -2+14=12 -4+7=3$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=-2 b=14$$

Tuhia anō te 4x^{2}+12x-7 hei \left(4x^{2}-2x\right)+\left(14x-7\right).

$$\left(4x^{2}-2x\right)+\left(14x-7\right)$$

Tauwehea te 2x i te tuatahi me te 7 i te rōpū tuarua.

$$2x\left(2x-1\right)+7\left(2x-1\right)$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(2x+7\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-1=0 me te 2x+7=0.

$$x=\frac{1}{2} x=-\frac{7}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 128 ki te x+1.

$$128+128x+128+128\left(x+1\right)^{2}=608$$

Tāpirihia te 128 ki te 128, ka 256.

$$256+128x+128\left(x+1\right)^{2}=608$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$256+128x+128\left(x^{2}+2x+1\right)=608$$

Whakamahia te āhuatanga tohatoha hei whakarea te 128 ki te x^{2}+2x+1.

$$256+128x+128x^{2}+256x+128=608$$

Pahekotia te 128x me 256x, ka 384x.

$$256+384x+128x^{2}+128=608$$

Tāpirihia te 256 ki te 128, ka 384.

$$384+384x+128x^{2}=608$$

Tangohia te 608 mai i ngā taha e rua.

$$384+384x+128x^{2}-608=0$$

Tangohia te 608 i te 384, ka -224.

$$-224+384x+128x^{2}=0$$

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.

$$128x^{2}+384x-224=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 128 mō a, 384 mō b, me -224 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-384±\sqrt{384^{2}-4\times 128\left(-224\right)}}{2\times 128}$$

Pūrua 384.

$$x=\frac{-384±\sqrt{147456-4\times 128\left(-224\right)}}{2\times 128}$$

Whakareatia -4 ki te 128.

$$x=\frac{-384±\sqrt{147456-512\left(-224\right)}}{2\times 128}$$

Whakareatia -512 ki te -224.

$$x=\frac{-384±\sqrt{147456+114688}}{2\times 128}$$

Tāpiri 147456 ki te 114688.

$$x=\frac{-384±\sqrt{262144}}{2\times 128}$$

Tuhia te pūtakerua o te 262144.

$$x=\frac{-384±512}{2\times 128}$$

Whakareatia 2 ki te 128.

$$x=\frac{-384±512}{256}$$

Nā, me whakaoti te whārite x=\frac{-384±512}{256} ina he tāpiri te ±. Tāpiri -384 ki te 512.

$$x=\frac{128}{256}$$

Whakahekea te hautanga \frac{128}{256} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 128.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-384±512}{256} ina he tango te ±. Tango 512 mai i -384.

$$x=-\frac{896}{256}$$

Whakahekea te hautanga \frac{-896}{256} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 128.

$$x=-\frac{7}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=-\frac{7}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 128 ki te x+1.

$$128+128x+128+128\left(x+1\right)^{2}=608$$

Tāpirihia te 128 ki te 128, ka 256.

$$256+128x+128\left(x+1\right)^{2}=608$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$256+128x+128\left(x^{2}+2x+1\right)=608$$

Whakamahia te āhuatanga tohatoha hei whakarea te 128 ki te x^{2}+2x+1.

$$256+128x+128x^{2}+256x+128=608$$

Pahekotia te 128x me 256x, ka 384x.

$$256+384x+128x^{2}+128=608$$

Tāpirihia te 256 ki te 128, ka 384.

$$384+384x+128x^{2}=608$$

Tangohia te 384 mai i ngā taha e rua.

$$384x+128x^{2}=608-384$$

Tangohia te 384 i te 608, ka 224.

$$384x+128x^{2}=224$$

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.

$$128x^{2}+384x=224$$

Whakawehea ngā taha e rua ki te 128.

$$\frac{128x^{2}+384x}{128}=\frac{224}{128}$$

Mā te whakawehe ki te 128 ka wetekia te whakareanga ki te 128.

$$x^{2}+\frac{384}{128}x=\frac{224}{128}$$

Whakawehe 384 ki te 128.

$$x^{2}+3x=\frac{224}{128}$$

Whakahekea te hautanga \frac{224}{128} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 32.

$$x^{2}+3x=\frac{7}{4}$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{7}{4}+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=\frac{7+9}{4}$$

Tāpiri \frac{7}{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.

$$x^{2}+3x+\frac{9}{4}=4$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=4$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

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

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{2} x=-\frac{7}{2}$$