Whakaoti i te 5x^2-25x=5x-40

$$5 x ^ { 2 } - 25 x = 5 x - 40$$

Whakaoti mō x

$$x=2$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 5x mai i ngā taha e rua.

$$5x^{2}-25x-5x=-40$$

Pahekotia te -25x me -5x, ka -30x.

$$5x^{2}-30x=-40$$

Me tāpiri te 40 ki ngā taha e rua.

$$5x^{2}-30x+40=0$$

Whakawehea ngā taha e rua ki te 5.

$$x^{2}-6x+8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 8=8$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 8.

$$-1,-8 -2,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-8=-9 -2-4=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-4 b=-2$$

Tuhia anō te x^{2}-6x+8 hei \left(x^{2}-4x\right)+\left(-2x+8\right).

$$\left(x^{2}-4x\right)+\left(-2x+8\right)$$

Tauwehea te x i te tuatahi me te -2 i te rōpū tuarua.

$$x\left(x-4\right)-2\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x-2=0.

$$x=4 x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 5x mai i ngā taha e rua.

$$5x^{2}-25x-5x=-40$$

Pahekotia te -25x me -5x, ka -30x.

$$5x^{2}-30x=-40$$

Me tāpiri te 40 ki ngā taha e rua.

$$5x^{2}-30x+40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -30 mō b, me 40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 5\times 40}}{2\times 5}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 5\times 40}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-30\right)±\sqrt{900-20\times 40}}{2\times 5}$$

Whakareatia -20 ki te 40.

$$x=\frac{-\left(-30\right)±\sqrt{900-800}}{2\times 5}$$

Tāpiri 900 ki te -800.

$$x=\frac{-\left(-30\right)±\sqrt{100}}{2\times 5}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-30\right)±10}{2\times 5}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±10}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{30±10}{10}$$

Nā, me whakaoti te whārite x=\frac{30±10}{10} ina he tāpiri te ±. Tāpiri 30 ki te 10.

$$x=\frac{40}{10}$$

Whakawehe 40 ki te 10.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{30±10}{10} ina he tango te ±. Tango 10 mai i 30.

$$x=\frac{20}{10}$$

Whakawehe 20 ki te 10.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=4 x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 5x mai i ngā taha e rua.

$$5x^{2}-25x-5x=-40$$

Pahekotia te -25x me -5x, ka -30x.

$$5x^{2}-30x=-40$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-30x}{5}=-\frac{40}{5}$$

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

$$x^{2}+\left(-\frac{30}{5}\right)x=-\frac{40}{5}$$

Whakawehe -30 ki te 5.

$$x^{2}-6x=-\frac{40}{5}$$

Whakawehe -40 ki te 5.

$$x^{2}-6x=-8$$

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

$$x^{2}-6x+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-8+9$$

Tāpiri -8 ki te 9.

$$x^{2}-6x+9=1$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=1$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-3=1 x-3=-1$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=4 x=2$$