Whakaoti i te 25x^2+250x-15000=0

$$25 { x }^{ 2 } +250x-15000=0$$

Whakaoti mō x

$$x=-30$$

$$x=20$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 25.

$$x^{2}+10x-600=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-600. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=1\left(-600\right)=-600$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -600.

$$-1,600 -2,300 -3,200 -4,150 -5,120 -6,100 -8,75 -10,60 -12,50 -15,40 -20,30 -24,25$$

Tātaihia te tapeke mō ia takirua.

$$-1+600=599 -2+300=298 -3+200=197 -4+150=146 -5+120=115 -6+100=94 -8+75=67 -10+60=50 -12+50=38 -15+40=25 -20+30=10 -24+25=1$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-20 b=30$$

Tuhia anō te x^{2}+10x-600 hei \left(x^{2}-20x\right)+\left(30x-600\right).

$$\left(x^{2}-20x\right)+\left(30x-600\right)$$

Tauwehea te x i te tuatahi me te 30 i te rōpū tuarua.

$$x\left(x-20\right)+30\left(x-20\right)$$

Whakatauwehea atu te kīanga pātahi x-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-20\right)\left(x+30\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x+30=0.

$$x=20 x=-30$$

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.

$$25x^{2}+250x-15000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, 250 mō b, me -15000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-250±\sqrt{250^{2}-4\times 25\left(-15000\right)}}{2\times 25}$$

Pūrua 250.

$$x=\frac{-250±\sqrt{62500-4\times 25\left(-15000\right)}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-250±\sqrt{62500-100\left(-15000\right)}}{2\times 25}$$

Whakareatia -100 ki te -15000.

$$x=\frac{-250±\sqrt{62500+1500000}}{2\times 25}$$

Tāpiri 62500 ki te 1500000.

$$x=\frac{-250±\sqrt{1562500}}{2\times 25}$$

Tuhia te pūtakerua o te 1562500.

$$x=\frac{-250±1250}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{-250±1250}{50}$$

Nā, me whakaoti te whārite x=\frac{-250±1250}{50} ina he tāpiri te ±. Tāpiri -250 ki te 1250.

$$x=\frac{1000}{50}$$

Whakawehe 1000 ki te 50.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{-250±1250}{50} ina he tango te ±. Tango 1250 mai i -250.

$$x=-\frac{1500}{50}$$

Whakawehe -1500 ki te 50.

$$x=-30$$

Kua oti te whārite te whakatau.

$$x=20 x=-30$$

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.

$$25x^{2}+250x-15000=0$$

Me tāpiri 15000 ki ngā taha e rua o te whārite.

$$25x^{2}+250x-15000-\left(-15000\right)=-\left(-15000\right)$$

Mā te tango i te -15000 i a ia ake anō ka toe ko te 0.

$$25x^{2}+250x=-\left(-15000\right)$$

Tango -15000 mai i 0.

$$25x^{2}+250x=15000$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}+250x}{25}=\frac{15000}{25}$$

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

$$x^{2}+\frac{250}{25}x=\frac{15000}{25}$$

Whakawehe 250 ki te 25.

$$x^{2}+10x=\frac{15000}{25}$$

Whakawehe 15000 ki te 25.

$$x^{2}+10x=600$$

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

$$x^{2}+10x+5^{2}=600+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=600+25$$

Tāpiri 600 ki te 25.

$$x^{2}+10x+25=625$$

Tauwehea x^{2}+10x+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}.

$$\left(x+5\right)^{2}=625$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{625}$$

Whakarūnātia.

$$x+5=25 x+5=-25$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x=20 x=-30$$