Whakaoti i te x^2-8x+10=3x

$${ x }^{ 2 } -8x+10 = 3x$$

Whakaoti mō x

$$x=1$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-8x+10-3x=0$$

Pahekotia te -8x me -3x, ka -11x.

$$x^{2}-11x+10=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-11x+10 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=10$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 10.

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

Tātaihia te tapeke mō ia takirua.

$$-1-10=-11 -2-5=-7$$

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

$$a=-10 b=-1$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-10\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x-1=0.

$$x=10 x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-8x+10-3x=0$$

Pahekotia te -8x me -3x, ka -11x.

$$x^{2}-11x+10=0$$

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.

$$a+b=-11 ab=1\times 10=10$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 10.

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

Tātaihia te tapeke mō ia takirua.

$$-1-10=-11 -2-5=-7$$

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

$$a=-10 b=-1$$

Tuhia anō te x^{2}-11x+10 hei \left(x^{2}-10x\right)+\left(-x+10\right).

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

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

$$x\left(x-10\right)-\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x-1=0.

$$x=10 x=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-8x+10-3x=0$$

Pahekotia te -8x me -3x, ka -11x.

$$x^{2}-11x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -11 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 10}}{2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 10}}{2}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-11\right)±\sqrt{121-40}}{2}$$

Tāpiri 121 ki te -40.

$$x=\frac{-\left(-11\right)±\sqrt{81}}{2}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-11\right)±9}{2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±9}{2}$$

Nā, me whakaoti te whārite x=\frac{11±9}{2} ina he tāpiri te ±. Tāpiri 11 ki te 9.

$$x=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{11±9}{2} ina he tango te ±. Tango 9 mai i 11.

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

Whakawehe 2 ki te 2.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=10 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-8x+10-3x=0$$

Pahekotia te -8x me -3x, ka -11x.

$$x^{2}-11x+10=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}-11x=-10$$

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

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-10+\left(-\frac{11}{2}\right)^{2}$$

Pūruatia -\frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-11x+\frac{121}{4}=-10+\frac{121}{4}$$

Tāpiri -10 ki te \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{81}{4}$$

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

$$\left(x-\frac{11}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

$$x-\frac{11}{2}=\frac{9}{2} x-\frac{11}{2}=-\frac{9}{2}$$

Me tāpiri \frac{11}{2} ki ngā taha e rua o te whārite.

$$x=10 x=1$$