Whakaoti i te 10x^2+10x+8=3x^2-10x+11

$$10 { x }^{ 2 } +10x+8=3 { x }^{ 2 } -10x+11$$

Whakaoti mō x

$$x=-3$$

$$x=\frac{1}{7}\approx 0.142857143$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3x^{2} mai i ngā taha e rua.

$$10x^{2}+10x+8-3x^{2}=-10x+11$$

Pahekotia te 10x^{2} me -3x^{2}, ka 7x^{2}.

$$7x^{2}+10x+8=-10x+11$$

Me tāpiri te 10x ki ngā taha e rua.

$$7x^{2}+10x+8+10x=11$$

Pahekotia te 10x me 10x, ka 20x.

$$7x^{2}+20x+8=11$$

Tangohia te 11 mai i ngā taha e rua.

$$7x^{2}+20x+8-11=0$$

Tangohia te 11 i te 8, ka -3.

$$7x^{2}+20x-3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 7x^{2}+ax+bx-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=7\left(-3\right)=-21$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -21.

$$-1,21 -3,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+21=20 -3+7=4$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=-1 b=21$$

Tuhia anō te 7x^{2}+20x-3 hei \left(7x^{2}-x\right)+\left(21x-3\right).

$$\left(7x^{2}-x\right)+\left(21x-3\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(7x-1\right)+3\left(7x-1\right)$$

Whakatauwehea atu te kīanga pātahi 7x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-1\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 7x-1=0 me te x+3=0.

$$x=\frac{1}{7} x=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3x^{2} mai i ngā taha e rua.

$$10x^{2}+10x+8-3x^{2}=-10x+11$$

Pahekotia te 10x^{2} me -3x^{2}, ka 7x^{2}.

$$7x^{2}+10x+8=-10x+11$$

Me tāpiri te 10x ki ngā taha e rua.

$$7x^{2}+10x+8+10x=11$$

Pahekotia te 10x me 10x, ka 20x.

$$7x^{2}+20x+8=11$$

Tangohia te 11 mai i ngā taha e rua.

$$7x^{2}+20x+8-11=0$$

Tangohia te 11 i te 8, ka -3.

$$7x^{2}+20x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 20 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 7\left(-3\right)}}{2\times 7}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\times 7\left(-3\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-20±\sqrt{400-28\left(-3\right)}}{2\times 7}$$

Whakareatia -28 ki te -3.

$$x=\frac{-20±\sqrt{400+84}}{2\times 7}$$

Tāpiri 400 ki te 84.

$$x=\frac{-20±\sqrt{484}}{2\times 7}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-20±22}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-20±22}{14}$$

Nā, me whakaoti te whārite x=\frac{-20±22}{14} ina he tāpiri te ±. Tāpiri -20 ki te 22.

$$x=\frac{2}{14}$$

Whakahekea te hautanga \frac{2}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{7}$$

Nā, me whakaoti te whārite x=\frac{-20±22}{14} ina he tango te ±. Tango 22 mai i -20.

$$x=-\frac{42}{14}$$

Whakawehe -42 ki te 14.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{7} x=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3x^{2} mai i ngā taha e rua.

$$10x^{2}+10x+8-3x^{2}=-10x+11$$

Pahekotia te 10x^{2} me -3x^{2}, ka 7x^{2}.

$$7x^{2}+10x+8=-10x+11$$

Me tāpiri te 10x ki ngā taha e rua.

$$7x^{2}+10x+8+10x=11$$

Pahekotia te 10x me 10x, ka 20x.

$$7x^{2}+20x+8=11$$

Tangohia te 8 mai i ngā taha e rua.

$$7x^{2}+20x=11-8$$

Tangohia te 8 i te 11, ka 3.

$$7x^{2}+20x=3$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}+20x}{7}=\frac{3}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}+\frac{20}{7}x=\frac{3}{7}$$

Whakawehea te \frac{20}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{10}{7}. Nā, tāpiria te pūrua o te \frac{10}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{20}{7}x+\left(\frac{10}{7}\right)^{2}=\frac{3}{7}+\left(\frac{10}{7}\right)^{2}$$

Pūruatia \frac{10}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{20}{7}x+\frac{100}{49}=\frac{3}{7}+\frac{100}{49}$$

Tāpiri \frac{3}{7} ki te \frac{100}{49} 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.

$$x^{2}+\frac{20}{7}x+\frac{100}{49}=\frac{121}{49}$$

Tauwehea te x^{2}+\frac{20}{7}x+\frac{100}{49}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{10}{7}\right)^{2}=\frac{121}{49}$$

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

$$\sqrt{\left(x+\frac{10}{7}\right)^{2}}=\sqrt{\frac{121}{49}}$$

Whakarūnātia.

$$x+\frac{10}{7}=\frac{11}{7} x+\frac{10}{7}=-\frac{11}{7}$$

Me tango \frac{10}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{7} x=-3$$