Whakaoti i te -x^2+4x-4=-x

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

Whakaoti mō x

$$x=1$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te x ki ngā taha e rua.

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

Pahekotia te 4x me x, ka 5x.

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

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

$$a+b=5 ab=-\left(-4\right)=4$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 4.

$$1,4 2,2$$

Tātaihia te tapeke mō ia takirua.

$$1+4=5 2+2=4$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=4 b=1$$

Tuhia anō te -x^{2}+5x-4 hei \left(-x^{2}+4x\right)+\left(x-4\right).

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

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

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

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

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

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

$$x=4 x=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te x ki ngā taha e rua.

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

Pahekotia te 4x me x, ka 5x.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 5 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-5±\sqrt{25+4\left(-4\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -4.

$$x=\frac{-5±\sqrt{25-16}}{2\left(-1\right)}$$

Tāpiri 25 ki te -16.

$$x=\frac{-5±\sqrt{9}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-5±3}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-5±3}{-2}$$

Nā, me whakaoti te whārite x=\frac{-5±3}{-2} ina he tāpiri te ±. Tāpiri -5 ki te 3.

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

Whakawehe -2 ki te -2.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-5±3}{-2} ina he tango te ±. Tango 3 mai i -5.

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

Whakawehe -8 ki te -2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=1 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te x ki ngā taha e rua.

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

Pahekotia te 4x me x, ka 5x.

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

Me tāpiri te 4 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+5x}{-1}=\frac{4}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{5}{-1}x=\frac{4}{-1}$$

Whakawehe 5 ki te -1.

$$x^{2}-5x=\frac{4}{-1}$$

Whakawehe 4 ki te -1.

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

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=-4+\frac{25}{4}$$

Tāpiri -4 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{9}{4}$$

Tauwehea x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{3}{2} x-\frac{5}{2}=-\frac{3}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=4 x=1$$