Whakaoti i te x^2-10x-600=0

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

Whakaoti mō x

$$x=-20$$

$$x=30$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-10x-600 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=-600$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$a=-30 b=20$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

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

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

$$x=30 x=-20$$

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 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 tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$a=-30 b=20$$

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

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

Tauwehea te x i te tuatahi me te 20 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-30 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=30 x=-20$$

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me -600 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-600\right)}}{2}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-600\right)}}{2}$$

Whakareatia -4 ki te -600.

$$x=\frac{-\left(-10\right)±\sqrt{100+2400}}{2}$$

Tāpiri 100 ki te 2400.

$$x=\frac{-\left(-10\right)±\sqrt{2500}}{2}$$

Tuhia te pūtakerua o te 2500.

$$x=\frac{-\left(-10\right)±50}{2}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±50}{2}$$

Nā, me whakaoti te whārite x=\frac{10±50}{2} ina he tāpiri te ±. Tāpiri 10 ki te 50.

$$x=\frac{60}{2}$$

Whakawehe 60 ki te 2.

$$x=30$$

Nā, me whakaoti te whārite x=\frac{10±50}{2} ina he tango te ±. Tango 50 mai i 10.

$$x=-\frac{40}{2}$$

Whakawehe -40 ki te 2.

$$x=-20$$

Kua oti te whārite te whakatau.

$$x=30 x=-20$$

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.

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

Me tāpiri 600 ki ngā taha e rua o te whārite.

$$x^{2}-10x-600-\left(-600\right)=-\left(-600\right)$$

Mā te tango i te -600 i a ia ake anō ka toe ko te 0.

$$x^{2}-10x=-\left(-600\right)$$

Tango -600 mai i 0.

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

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.

$$x^{2}-10x+\left(-5\right)^{2}=600+\left(-5\right)^{2}$$

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 tāpiri 5 ki ngā taha e rua o te whārite.

$$x=30 x=-20$$