Whakaoti i te x^2-2=11x-5/4

$$x ^ { 2 } - 2 = \frac { 11 x - 5 } { 4 }$$

Whakaoti mō x

$$x=-\frac{1}{4}=-0.25$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4x^{2}-8=11x-5$$

Tangohia te 11x mai i ngā taha e rua.

$$4x^{2}-8-11x=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$4x^{2}-8-11x+5=0$$

Tāpirihia te -8 ki te 5, ka -3.

$$4x^{2}-3-11x=0$$

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

$$4x^{2}-11x-3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=4\left(-3\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-12 b=1$$

Tuhia anō te 4x^{2}-11x-3 hei \left(4x^{2}-12x\right)+\left(x-3\right).

$$\left(4x^{2}-12x\right)+\left(x-3\right)$$

Whakatauwehea atu 4x i te 4x^{2}-12x.

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

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

$$\left(x-3\right)\left(4x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te 4x+1=0.

$$x=3 x=-\frac{1}{4}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4x^{2}-8=11x-5$$

Tangohia te 11x mai i ngā taha e rua.

$$4x^{2}-8-11x=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$4x^{2}-8-11x+5=0$$

Tāpirihia te -8 ki te 5, ka -3.

$$4x^{2}-3-11x=0$$

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.

$$4x^{2}-11x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -11 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\left(-3\right)}}{2\times 4}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\left(-3\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\left(-3\right)}}{2\times 4}$$

Whakareatia -16 ki te -3.

$$x=\frac{-\left(-11\right)±\sqrt{121+48}}{2\times 4}$$

Tāpiri 121 ki te 48.

$$x=\frac{-\left(-11\right)±\sqrt{169}}{2\times 4}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-11\right)±13}{2\times 4}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±13}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{11±13}{8}$$

Nā, me whakaoti te whārite x=\frac{11±13}{8} ina he tāpiri te ±. Tāpiri 11 ki te 13.

$$x=\frac{24}{8}$$

Whakawehe 24 ki te 8.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{11±13}{8} ina he tango te ±. Tango 13 mai i 11.

$$x=-\frac{2}{8}$$

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

$$x=-\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$x=3 x=-\frac{1}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4x^{2}-8=11x-5$$

Tangohia te 11x mai i ngā taha e rua.

$$4x^{2}-8-11x=-5$$

Me tāpiri te 8 ki ngā taha e rua.

$$4x^{2}-11x=-5+8$$

Tāpirihia te -5 ki te 8, ka 3.

$$4x^{2}-11x=3$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-11x}{4}=\frac{3}{4}$$

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

$$x^{2}-\frac{11}{4}x=\frac{3}{4}$$

Whakawehea te -\frac{11}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{8}. Nā, tāpiria te pūrua o te -\frac{11}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=\frac{3}{4}+\left(-\frac{11}{8}\right)^{2}$$

Pūruatia -\frac{11}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=\frac{3}{4}+\frac{121}{64}$$

Tāpiri \frac{3}{4} ki te \frac{121}{64} 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.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=\frac{169}{64}$$

Tauwehea x^{2}-\frac{11}{4}x+\frac{121}{64}. 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}.

$$\left(x-\frac{11}{8}\right)^{2}=\frac{169}{64}$$

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

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Whakarūnātia.

$$x-\frac{11}{8}=\frac{13}{8} x-\frac{11}{8}=-\frac{13}{8}$$

Me tāpiri \frac{11}{8} ki ngā taha e rua o te whārite.

$$x=3 x=-\frac{1}{4}$$