Whakaoti i te 5x^2-20x+15=0

$$5 x ^ { 2 } - 20 x + 15 = 0$$

Whakaoti mō x

$$x=1$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 5.

$$x^{2}-4x+3=0$$

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.

$$a+b=-4 ab=1\times 3=3$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-3 b=-1$$

Tuhia anō te x^{2}-4x+3 hei \left(x^{2}-3x\right)+\left(-x+3\right).

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

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

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

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.

$$5x^{2}-20x+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -20 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 5\times 15}}{2\times 5}$$

Pūrua -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 5\times 15}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-20\right)±\sqrt{400-20\times 15}}{2\times 5}$$

Whakareatia -20 ki te 15.

$$x=\frac{-\left(-20\right)±\sqrt{400-300}}{2\times 5}$$

Tāpiri 400 ki te -300.

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

Tuhia te pūtakerua o te 100.

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

Ko te tauaro o -20 ko 20.

$$x=\frac{20±10}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{20±10}{10}$$

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

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

Whakawehe 30 ki te 10.

$$x=3$$

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

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

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.

$$5x^{2}-20x+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$5x^{2}-20x+15-15=-15$$

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

$$5x^{2}-20x=-15$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-20x}{5}=-\frac{15}{5}$$

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

$$x^{2}+\left(-\frac{20}{5}\right)x=-\frac{15}{5}$$

Whakawehe -20 ki te 5.

$$x^{2}-4x=-\frac{15}{5}$$

Whakawehe -15 ki te 5.

$$x^{2}-4x=-3$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-4x+\left(-2\right)^{2}=-3+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=-3+4$$

Tāpiri -3 ki te 4.

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

Tauwehea x^{2}-4x+4. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=3 x=1$$