Whakaoti i te 10x^2-11x+9=13x-6x^2

$$10 { x }^{ 2 } -11x+9=13x-6 { x }^{ 2 }$$

Whakaoti mō x

$$x=\frac{3}{4}=0.75$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 13x mai i ngā taha e rua.

$$10x^{2}-11x+9-13x=-6x^{2}$$

Pahekotia te -11x me -13x, ka -24x.

$$10x^{2}-24x+9=-6x^{2}$$

Me tāpiri te 6x^{2} ki ngā taha e rua.

$$10x^{2}-24x+9+6x^{2}=0$$

Pahekotia te 10x^{2} me 6x^{2}, ka 16x^{2}.

$$16x^{2}-24x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 16x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-24 ab=16\times 9=144$$

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 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

Ko te otinga te takirua ka hoatu i te tapeke -24.

$$a=-12 b=-12$$

Tuhia anō te 16x^{2}-24x+9 hei \left(16x^{2}-12x\right)+\left(-12x+9\right).

$$\left(16x^{2}-12x\right)+\left(-12x+9\right)$$

Tauwehea te 4x i te tuatahi me te -3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 4x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te 4x-3=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 13x mai i ngā taha e rua.

$$10x^{2}-11x+9-13x=-6x^{2}$$

Pahekotia te -11x me -13x, ka -24x.

$$10x^{2}-24x+9=-6x^{2}$$

Me tāpiri te 6x^{2} ki ngā taha e rua.

$$10x^{2}-24x+9+6x^{2}=0$$

Pahekotia te 10x^{2} me 6x^{2}, ka 16x^{2}.

$$16x^{2}-24x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -24 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 16\times 9}}{2\times 16}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 16\times 9}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-24\right)±\sqrt{576-64\times 9}}{2\times 16}$$

Whakareatia -64 ki te 9.

$$x=\frac{-\left(-24\right)±\sqrt{576-576}}{2\times 16}$$

Tāpiri 576 ki te -576.

$$x=\frac{-\left(-24\right)±\sqrt{0}}{2\times 16}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-24}{2\times 16}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{24}{32}$$

Whakahekea te hautanga \frac{24}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 13x mai i ngā taha e rua.

$$10x^{2}-11x+9-13x=-6x^{2}$$

Pahekotia te -11x me -13x, ka -24x.

$$10x^{2}-24x+9=-6x^{2}$$

Me tāpiri te 6x^{2} ki ngā taha e rua.

$$10x^{2}-24x+9+6x^{2}=0$$

Pahekotia te 10x^{2} me 6x^{2}, ka 16x^{2}.

$$16x^{2}-24x+9=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$16x^{2}-24x=-9$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}-24x}{16}=-\frac{9}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\left(-\frac{24}{16}\right)x=-\frac{9}{16}$$

Whakahekea te hautanga \frac{-24}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x^{2}-\frac{3}{2}x=-\frac{9}{16}$$

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{9}{16}+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{-9+9}{16}$$

Tāpiri -\frac{9}{16} ki te \frac{9}{16} 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.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=0$$

Tauwehea x^{2}-\frac{3}{2}x+\frac{9}{16}. 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}.

$$\left(x-\frac{3}{4}\right)^{2}=0$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-\frac{3}{4}=0 x-\frac{3}{4}=0$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

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

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

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