Whakaoti i te 3x(x-5)+16=-x

$$3x(x-5)+16=-x$$

Whakaoti mō x

$$x=2$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te x-5.

$$3x^{2}-15x+16=-x$$

Me tāpiri te x ki ngā taha e rua.

$$3x^{2}-15x+16+x=0$$

Pahekotia te -15x me x, ka -14x.

$$3x^{2}-14x+16=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx+16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=3\times 16=48$$

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 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-8 b=-6$$

Tuhia anō te 3x^{2}-14x+16 hei \left(3x^{2}-8x\right)+\left(-6x+16\right).

$$\left(3x^{2}-8x\right)+\left(-6x+16\right)$$

Tauwehea te x i te tuatahi me te -2 i te rōpū tuarua.

$$x\left(3x-8\right)-2\left(3x-8\right)$$

Whakatauwehea atu te kīanga pātahi 3x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-8\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-8=0 me te x-2=0.

$$x=\frac{8}{3} x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te x-5.

$$3x^{2}-15x+16=-x$$

Me tāpiri te x ki ngā taha e rua.

$$3x^{2}-15x+16+x=0$$

Pahekotia te -15x me x, ka -14x.

$$3x^{2}-14x+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -14 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 3\times 16}}{2\times 3}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 3\times 16}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-14\right)±\sqrt{196-12\times 16}}{2\times 3}$$

Whakareatia -12 ki te 16.

$$x=\frac{-\left(-14\right)±\sqrt{196-192}}{2\times 3}$$

Tāpiri 196 ki te -192.

$$x=\frac{-\left(-14\right)±\sqrt{4}}{2\times 3}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-\left(-14\right)±2}{2\times 3}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±2}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{14±2}{6}$$

Nā, me whakaoti te whārite x=\frac{14±2}{6} ina he tāpiri te ±. Tāpiri 14 ki te 2.

$$x=\frac{16}{6}$$

Whakahekea te hautanga \frac{16}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{8}{3}$$

Nā, me whakaoti te whārite x=\frac{14±2}{6} ina he tango te ±. Tango 2 mai i 14.

$$x=\frac{12}{6}$$

Whakawehe 12 ki te 6.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=\frac{8}{3} x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te x-5.

$$3x^{2}-15x+16=-x$$

Me tāpiri te x ki ngā taha e rua.

$$3x^{2}-15x+16+x=0$$

Pahekotia te -15x me x, ka -14x.

$$3x^{2}-14x+16=0$$

Tangohia te 16 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x^{2}-14x=-16$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-14x}{3}=-\frac{16}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{14}{3}x=-\frac{16}{3}$$

Whakawehea te -\frac{14}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{3}. Nā, tāpiria te pūrua o te -\frac{7}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=-\frac{16}{3}+\left(-\frac{7}{3}\right)^{2}$$

Pūruatia -\frac{7}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-\frac{16}{3}+\frac{49}{9}$$

Tāpiri -\frac{16}{3} ki te \frac{49}{9} 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.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{1}{9}$$

Tauwehea x^{2}-\frac{14}{3}x+\frac{49}{9}. 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}.

$$\left(x-\frac{7}{3}\right)^{2}=\frac{1}{9}$$

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

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Whakarūnātia.

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

Me tāpiri \frac{7}{3} ki ngā taha e rua o te whārite.

$$x=\frac{8}{3} x=2$$