Whakaoti i te left(2x-3right)^2=49

$${ \left(2x-3 \right) }^{ 2 } =49$$

Whakaoti mō x

$$x=5$$

$$x=-2$$

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-3\right)^{2}.

$$4x^{2}-12x+9=49$$

Tangohia te 49 mai i ngā taha e rua.

$$4x^{2}-12x+9-49=0$$

Tangohia te 49 i te 9, ka -40.

$$4x^{2}-12x-40=0$$

Whakawehea ngā taha e rua ki te 4.

$$x^{2}-3x-10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-10\right)=-10$$

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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-5 b=2$$

Tuhia anō te x^{2}-3x-10 hei \left(x^{2}-5x\right)+\left(2x-10\right).

$$\left(x^{2}-5x\right)+\left(2x-10\right)$$

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

$$x\left(x-5\right)+2\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+2=0.

$$x=5 x=-2$$

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-3\right)^{2}.

$$4x^{2}-12x+9=49$$

Tangohia te 49 mai i ngā taha e rua.

$$4x^{2}-12x+9-49=0$$

Tangohia te 49 i te 9, ka -40.

$$4x^{2}-12x-40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -12 mō b, me -40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-40\right)}}{2\times 4}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-40\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-40\right)}}{2\times 4}$$

Whakareatia -16 ki te -40.

$$x=\frac{-\left(-12\right)±\sqrt{144+640}}{2\times 4}$$

Tāpiri 144 ki te 640.

$$x=\frac{-\left(-12\right)±\sqrt{784}}{2\times 4}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-\left(-12\right)±28}{2\times 4}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±28}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{12±28}{8}$$

Nā, me whakaoti te whārite x=\frac{12±28}{8} ina he tāpiri te ±. Tāpiri 12 ki te 28.

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

Whakawehe 40 ki te 8.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{12±28}{8} ina he tango te ±. Tango 28 mai i 12.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=5 x=-2$$

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-3\right)^{2}.

$$4x^{2}-12x+9=49$$

Tangohia te 9 mai i ngā taha e rua.

$$4x^{2}-12x=49-9$$

Tangohia te 9 i te 49, ka 40.

$$4x^{2}-12x=40$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-12x}{4}=\frac{40}{4}$$

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

$$x^{2}+\left(-\frac{12}{4}\right)x=\frac{40}{4}$$

Whakawehe -12 ki te 4.

$$x^{2}-3x=\frac{40}{4}$$

Whakawehe 40 ki te 4.

$$x^{2}-3x=10$$

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.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=10+\left(-\frac{3}{2}\right)^{2}$$

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

$$x^{2}-3x+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{49}{4}$$

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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=5 x=-2$$