Whakaoti i te (2x+1)^2=(x-5)^2

$$( 2 x + 1 ) ^ { 2 } = ( x - 5 ) ^ { 2 }$$

Whakaoti mō x

$$x=-6$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$4x^{2}+4x+1=\left(x-5\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-5\right)^{2}.

$$4x^{2}+4x+1=x^{2}-10x+25$$

Tangohia te x^{2} mai i ngā taha e rua.

$$4x^{2}+4x+1-x^{2}=-10x+25$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}+4x+1=-10x+25$$

Me tāpiri te 10x ki ngā taha e rua.

$$3x^{2}+4x+1+10x=25$$

Pahekotia te 4x me 10x, ka 14x.

$$3x^{2}+14x+1=25$$

Tangohia te 25 mai i ngā taha e rua.

$$3x^{2}+14x+1-25=0$$

Tangohia te 25 i te 1, ka -24.

$$3x^{2}+14x-24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=3\left(-24\right)=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=-4 b=18$$

Tuhia anō te 3x^{2}+14x-24 hei \left(3x^{2}-4x\right)+\left(18x-24\right).

$$\left(3x^{2}-4x\right)+\left(18x-24\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(3x-4\right)+6\left(3x-4\right)$$

Whakatauwehea atu te kīanga pātahi 3x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-4\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-4=0 me te x+6=0.

$$x=\frac{4}{3} x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$4x^{2}+4x+1=\left(x-5\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-5\right)^{2}.

$$4x^{2}+4x+1=x^{2}-10x+25$$

Tangohia te x^{2} mai i ngā taha e rua.

$$4x^{2}+4x+1-x^{2}=-10x+25$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}+4x+1=-10x+25$$

Me tāpiri te 10x ki ngā taha e rua.

$$3x^{2}+4x+1+10x=25$$

Pahekotia te 4x me 10x, ka 14x.

$$3x^{2}+14x+1=25$$

Tangohia te 25 mai i ngā taha e rua.

$$3x^{2}+14x+1-25=0$$

Tangohia te 25 i te 1, ka -24.

$$3x^{2}+14x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 14 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 3\left(-24\right)}}{2\times 3}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 3\left(-24\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-14±\sqrt{196-12\left(-24\right)}}{2\times 3}$$

Whakareatia -12 ki te -24.

$$x=\frac{-14±\sqrt{196+288}}{2\times 3}$$

Tāpiri 196 ki te 288.

$$x=\frac{-14±\sqrt{484}}{2\times 3}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-14±22}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-14±22}{6}$$

Nā, me whakaoti te whārite x=\frac{-14±22}{6} ina he tāpiri te ±. Tāpiri -14 ki te 22.

$$x=\frac{8}{6}$$

Whakahekea te hautanga \frac{8}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{4}{3}$$

Nā, me whakaoti te whārite x=\frac{-14±22}{6} ina he tango te ±. Tango 22 mai i -14.

$$x=-\frac{36}{6}$$

Whakawehe -36 ki te 6.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=\frac{4}{3} x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$4x^{2}+4x+1=\left(x-5\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-5\right)^{2}.

$$4x^{2}+4x+1=x^{2}-10x+25$$

Tangohia te x^{2} mai i ngā taha e rua.

$$4x^{2}+4x+1-x^{2}=-10x+25$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}+4x+1=-10x+25$$

Me tāpiri te 10x ki ngā taha e rua.

$$3x^{2}+4x+1+10x=25$$

Pahekotia te 4x me 10x, ka 14x.

$$3x^{2}+14x+1=25$$

Tangohia te 1 mai i ngā taha e rua.

$$3x^{2}+14x=25-1$$

Tangohia te 1 i te 25, ka 24.

$$3x^{2}+14x=24$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+14x}{3}=\frac{24}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{14}{3}x=\frac{24}{3}$$

Whakawehe 24 ki te 3.

$$x^{2}+\frac{14}{3}x=8$$

Whakawehea te \frac{14}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{3}. Nā, tāpiria te pūrua o te \frac{7}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{14}{3}x+\left(\frac{7}{3}\right)^{2}=8+\left(\frac{7}{3}\right)^{2}$$

Pūruatia \frac{7}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=8+\frac{49}{9}$$

Tāpiri 8 ki te \frac{49}{9}.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=\frac{121}{9}$$

Tauwehea x^{2}+\frac{14}{3}x+\frac{49}{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}.

$$\left(x+\frac{7}{3}\right)^{2}=\frac{121}{9}$$

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

$$\sqrt{\left(x+\frac{7}{3}\right)^{2}}=\sqrt{\frac{121}{9}}$$

Whakarūnātia.

$$x+\frac{7}{3}=\frac{11}{3} x+\frac{7}{3}=-\frac{11}{3}$$

Me tango \frac{7}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{4}{3} x=-6$$