Whakaoti i te 3x^2+45=24x

$$3 x ^ { 2 } + 45 = 24 x$$

Whakaoti mō x

$$x=3$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 24x mai i ngā taha e rua.

$$3x^{2}+45-24x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x^{2}+15-8x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-8x+15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 15=15$$

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 15.

$$-1,-15 -3,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-15=-16 -3-5=-8$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-5 b=-3$$

Tuhia anō te x^{2}-8x+15 hei \left(x^{2}-5x\right)+\left(-3x+15\right).

$$\left(x^{2}-5x\right)+\left(-3x+15\right)$$

Tauwehea te x i te tuatahi me te -3 i te rōpū tuarua.

$$x\left(x-5\right)-3\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x-3=0.

$$x=5 x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 24x mai i ngā taha e rua.

$$3x^{2}+45-24x=0$$

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.

$$3x^{2}-24x+45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -24 mō b, me 45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 3\times 45}}{2\times 3}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 3\times 45}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-24\right)±\sqrt{576-12\times 45}}{2\times 3}$$

Whakareatia -12 ki te 45.

$$x=\frac{-\left(-24\right)±\sqrt{576-540}}{2\times 3}$$

Tāpiri 576 ki te -540.

$$x=\frac{-\left(-24\right)±\sqrt{36}}{2\times 3}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-24\right)±6}{2\times 3}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±6}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{24±6}{6}$$

Nā, me whakaoti te whārite x=\frac{24±6}{6} ina he tāpiri te ±. Tāpiri 24 ki te 6.

$$x=\frac{30}{6}$$

Whakawehe 30 ki te 6.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{24±6}{6} ina he tango te ±. Tango 6 mai i 24.

$$x=\frac{18}{6}$$

Whakawehe 18 ki te 6.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=5 x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 24x mai i ngā taha e rua.

$$3x^{2}+45-24x=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x^{2}-24x=-45$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-24x}{3}=-\frac{45}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\left(-\frac{24}{3}\right)x=-\frac{45}{3}$$

Whakawehe -24 ki te 3.

$$x^{2}-8x=-\frac{45}{3}$$

Whakawehe -45 ki te 3.

$$x^{2}-8x=-15$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=-15+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-15+16$$

Tāpiri -15 ki te 16.

$$x^{2}-8x+16=1$$

Tauwehea te x^{2}-8x+16. 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}.

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

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

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

Whakarūnātia.

$$x-4=1 x-4=-1$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=5 x=3$$