Whakaoti i te 9x^2-30x+25=0

$$9 { x }^{ 2 } -30x+25=0$$

Whakaoti mō x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 9x^{2}+ax+bx+25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-30 ab=9\times 25=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Tātaihia te tapeke mō ia takirua.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

Ko te otinga te takirua ka hoatu i te tapeke -30.

$$a=-15 b=-15$$

Tuhia anō te 9x^{2}-30x+25 hei \left(9x^{2}-15x\right)+\left(-15x+25\right).

$$\left(9x^{2}-15x\right)+\left(-15x+25\right)$$

Tauwehea te 3x i te tuatahi me te -5 i te rōpū tuarua.

$$3x\left(3x-5\right)-5\left(3x-5\right)$$

Whakatauwehea atu te kīanga pātahi 3x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-5\right)\left(3x-5\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(3x-5\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te 3x-5=0.

$$x=\frac{5}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$9x^{2}-30x+25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -30 mō b, me 25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 9\times 25}}{2\times 9}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 9\times 25}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-30\right)±\sqrt{900-36\times 25}}{2\times 9}$$

Whakareatia -36 ki te 25.

$$x=\frac{-\left(-30\right)±\sqrt{900-900}}{2\times 9}$$

Tāpiri 900 ki te -900.

$$x=\frac{-\left(-30\right)±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-30}{2\times 9}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{30}{18}$$

Whakahekea te hautanga \frac{30}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{5}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$9x^{2}-30x+25=0$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$9x^{2}-30x+25-25=-25$$

Mā te tango i te 25 i a ia ake anō ka toe ko te 0.

$$9x^{2}-30x=-25$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-30x}{9}=-\frac{25}{9}$$

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

$$x^{2}+\left(-\frac{30}{9}\right)x=-\frac{25}{9}$$

Whakahekea te hautanga \frac{-30}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{10}{3}x=-\frac{25}{9}$$

Whakawehea te -\frac{10}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{3}. Nā, tāpiria te pūrua o te -\frac{5}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{10}{3}x+\left(-\frac{5}{3}\right)^{2}=-\frac{25}{9}+\left(-\frac{5}{3}\right)^{2}$$

Pūruatia -\frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=\frac{-25+25}{9}$$

Tāpiri -\frac{25}{9} ki te \frac{25}{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.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=0$$

Tauwehea te x^{2}-\frac{10}{3}x+\frac{25}{9}. 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}.

$$\left(x-\frac{5}{3}\right)^{2}=0$$

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

$$\sqrt{\left(x-\frac{5}{3}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-\frac{5}{3}=0 x-\frac{5}{3}=0$$

Me tāpiri \frac{5}{3} ki ngā taha e rua o te whārite.

$$x=\frac{5}{3} x=\frac{5}{3}$$

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

$$x=\frac{5}{3}$$