Whakaoti i te (2x-1)^2=121

$$( 2 x - 1 ) ^ { 2 } = 121$$

Whakaoti mō x

$$x=6$$

$$x=-5$$

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}.

$$4x^{2}-4x+1=121$$

Tangohia te 121 mai i ngā taha e rua.

$$4x^{2}-4x+1-121=0$$

Tangohia te 121 i te 1, ka -120.

$$4x^{2}-4x-120=0$$

Whakawehea ngā taha e rua ki te 4.

$$x^{2}-x-30=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-30\right)=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-6 b=5$$

Tuhia anō te x^{2}-x-30 hei \left(x^{2}-6x\right)+\left(5x-30\right).

$$\left(x^{2}-6x\right)+\left(5x-30\right)$$

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

$$x\left(x-6\right)+5\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x+5=0.

$$x=6 x=-5$$

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}.

$$4x^{2}-4x+1=121$$

Tangohia te 121 mai i ngā taha e rua.

$$4x^{2}-4x+1-121=0$$

Tangohia te 121 i te 1, ka -120.

$$4x^{2}-4x-120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -4 mō b, me -120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-120\right)}}{2\times 4}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-120\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-120\right)}}{2\times 4}$$

Whakareatia -16 ki te -120.

$$x=\frac{-\left(-4\right)±\sqrt{16+1920}}{2\times 4}$$

Tāpiri 16 ki te 1920.

$$x=\frac{-\left(-4\right)±\sqrt{1936}}{2\times 4}$$

Tuhia te pūtakerua o te 1936.

$$x=\frac{-\left(-4\right)±44}{2\times 4}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±44}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{4±44}{8}$$

Nā, me whakaoti te whārite x=\frac{4±44}{8} ina he tāpiri te ±. Tāpiri 4 ki te 44.

$$x=\frac{48}{8}$$

Whakawehe 48 ki te 8.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{4±44}{8} ina he tango te ±. Tango 44 mai i 4.

$$x=-\frac{40}{8}$$

Whakawehe -40 ki te 8.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=6 x=-5$$

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}.

$$4x^{2}-4x+1=121$$

Tangohia te 1 mai i ngā taha e rua.

$$4x^{2}-4x=121-1$$

Tangohia te 1 i te 121, ka 120.

$$4x^{2}-4x=120$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-4x}{4}=\frac{120}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{120}{4}$$

Whakawehe -4 ki te 4.

$$x^{2}-x=\frac{120}{4}$$

Whakawehe 120 ki te 4.

$$x^{2}-x=30$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=30+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=30+\frac{1}{4}$$

Tāpiri 30 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{121}{4}$$

Tauwehea x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{11}{2} x-\frac{1}{2}=-\frac{11}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=6 x=-5$$