Whakaoti i te 5x^2-10x-15=0

$$5 { x }^{ 2 } -10x-15=0$$

Whakaoti mō x

$$x=-1$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 5.

$$x^{2}-2x-3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-3\right)=-3$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-3 b=1$$

Tuhia anō te x^{2}-2x-3 hei \left(x^{2}-3x\right)+\left(x-3\right).

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

Whakatauwehea atu x i te x^{2}-3x.

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

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

$$\left(x-3\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te x+1=0.

$$x=3 x=-1$$

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.

$$5x^{2}-10x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -10 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 5\left(-15\right)}}{2\times 5}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 5\left(-15\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-10\right)±\sqrt{100-20\left(-15\right)}}{2\times 5}$$

Whakareatia -20 ki te -15.

$$x=\frac{-\left(-10\right)±\sqrt{100+300}}{2\times 5}$$

Tāpiri 100 ki te 300.

$$x=\frac{-\left(-10\right)±\sqrt{400}}{2\times 5}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-10\right)±20}{2\times 5}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±20}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{10±20}{10}$$

Nā, me whakaoti te whārite x=\frac{10±20}{10} ina he tāpiri te ±. Tāpiri 10 ki te 20.

$$x=\frac{30}{10}$$

Whakawehe 30 ki te 10.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{10±20}{10} ina he tango te ±. Tango 20 mai i 10.

$$x=-\frac{10}{10}$$

Whakawehe -10 ki te 10.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=3 x=-1$$

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.

$$5x^{2}-10x-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$5x^{2}-10x-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$5x^{2}-10x=-\left(-15\right)$$

Tango -15 mai i 0.

$$5x^{2}-10x=15$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-10x}{5}=\frac{15}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{10}{5}\right)x=\frac{15}{5}$$

Whakawehe -10 ki te 5.

$$x^{2}-2x=\frac{15}{5}$$

Whakawehe 15 ki te 5.

$$x^{2}-2x=3$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=3+1$$

Tāpiri 3 ki te 1.

$$x^{2}-2x+1=4$$

Tauwehea te x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=4$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-1=2 x-1=-2$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=3 x=-1$$