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

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

Whakaoti mō x

$$x=-\frac{3}{5}=-0.6$$

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 25x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga 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 25x^{2}+30x+9 hei \left(25x^{2}+15x\right)+\left(15x+9\right).

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

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

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

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

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

Tuhia anōtia hei pūrua huarua.

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

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

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

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.

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

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

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

Pūrua 30.

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

Whakareatia -4 ki te 25.

$$x=\frac{-30±\sqrt{900-100\times 9}}{2\times 25}$$

Whakareatia -100 ki te 9.

$$x=\frac{-30±\sqrt{900-900}}{2\times 25}$$

Tāpiri 900 ki te -900.

$$x=\frac{-30±\sqrt{0}}{2\times 25}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{30}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=-\frac{30}{50}$$

Whakahekea te hautanga \frac{-30}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

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.

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

Me tango 9 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te 25.

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

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

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

Whakahekea te hautanga \frac{30}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{6}{5}x=-\frac{9}{25}$$

Whakawehea te \frac{6}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{5}. Nā, tāpiria te pūrua o te \frac{3}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=-\frac{9}{25}+\left(\frac{3}{5}\right)^{2}$$

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

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{-9+9}{25}$$

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

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=0$$

Tauwehea x^{2}+\frac{6}{5}x+\frac{9}{25}. 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}.

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

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

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

Whakarūnātia.

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

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

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

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

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