Whakaoti i te left(3xright)^2-13x+4=0

$${ \left(3x \right) }^{ 2 } -13x+4 = 0$$

Whakaoti mō x

$$x=1$$

$$x=\frac{4}{9}\approx 0.444444444$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakarohaina te \left(3x\right)^{2}.

$$3^{2}x^{2}-13x+4=0$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$9x^{2}-13x+4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 9x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-13 ab=9\times 4=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=-9 b=-4$$

Tuhia anō te 9x^{2}-13x+4 hei \left(9x^{2}-9x\right)+\left(-4x+4\right).

$$\left(9x^{2}-9x\right)+\left(-4x+4\right)$$

Tauwehea te 9x i te tuatahi me te -4 i te rōpū tuarua.

$$9x\left(x-1\right)-4\left(x-1\right)$$

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

$$\left(x-1\right)\left(9x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 9x-4=0.

$$x=1 x=\frac{4}{9}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakarohaina te \left(3x\right)^{2}.

$$3^{2}x^{2}-13x+4=0$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$9x^{2}-13x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -13 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 9\times 4}}{2\times 9}$$

Pūrua -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 9\times 4}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-13\right)±\sqrt{169-36\times 4}}{2\times 9}$$

Whakareatia -36 ki te 4.

$$x=\frac{-\left(-13\right)±\sqrt{169-144}}{2\times 9}$$

Tāpiri 169 ki te -144.

$$x=\frac{-\left(-13\right)±\sqrt{25}}{2\times 9}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-13\right)±5}{2\times 9}$$

Ko te tauaro o -13 ko 13.

$$x=\frac{13±5}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{13±5}{18}$$

Nā, me whakaoti te whārite x=\frac{13±5}{18} ina he tāpiri te ±. Tāpiri 13 ki te 5.

$$x=\frac{18}{18}$$

Whakawehe 18 ki te 18.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{13±5}{18} ina he tango te ±. Tango 5 mai i 13.

$$x=\frac{8}{18}$$

Whakahekea te hautanga \frac{8}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{4}{9}$$

Kua oti te whārite te whakatau.

$$x=1 x=\frac{4}{9}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakarohaina te \left(3x\right)^{2}.

$$3^{2}x^{2}-13x+4=0$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$9x^{2}-13x+4=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9x^{2}-13x=-4$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-13x}{9}=-\frac{4}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}-\frac{13}{9}x=-\frac{4}{9}$$

Whakawehea te -\frac{13}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{18}. Nā, tāpiria te pūrua o te -\frac{13}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{9}x+\left(-\frac{13}{18}\right)^{2}=-\frac{4}{9}+\left(-\frac{13}{18}\right)^{2}$$

Pūruatia -\frac{13}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{9}x+\frac{169}{324}=-\frac{4}{9}+\frac{169}{324}$$

Tāpiri -\frac{4}{9} ki te \frac{169}{324} 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.

$$x^{2}-\frac{13}{9}x+\frac{169}{324}=\frac{25}{324}$$

Tauwehea x^{2}-\frac{13}{9}x+\frac{169}{324}. 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}.

$$\left(x-\frac{13}{18}\right)^{2}=\frac{25}{324}$$

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

$$\sqrt{\left(x-\frac{13}{18}\right)^{2}}=\sqrt{\frac{25}{324}}$$

Whakarūnātia.

$$x-\frac{13}{18}=\frac{5}{18} x-\frac{13}{18}=-\frac{5}{18}$$

Me tāpiri \frac{13}{18} ki ngā taha e rua o te whārite.

$$x=1 x=\frac{4}{9}$$