Whakaoti i te x^2+80=24x

$${ x }^{ 2 } +80 = 24x$$

Whakaoti mō x

$$x=4$$

$$x=20$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 24x mai i ngā taha e rua.

$$x^{2}+80-24x=0$$

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

$$x^{2}-24x+80=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-24x+80 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.

$$a+b=-24 ab=80$$

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 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

Ko te otinga te takirua ka hoatu i te tapeke -24.

$$a=-20 b=-4$$

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

$$\left(x-20\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x-4=0.

$$x=20 x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 24x mai i ngā taha e rua.

$$x^{2}+80-24x=0$$

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

$$x^{2}-24x+80=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+80. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-24 ab=1\times 80=80$$

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 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

Ko te otinga te takirua ka hoatu i te tapeke -24.

$$a=-20 b=-4$$

Tuhia anō te x^{2}-24x+80 hei \left(x^{2}-20x\right)+\left(-4x+80\right).

$$\left(x^{2}-20x\right)+\left(-4x+80\right)$$

Tauwehea te x i te tuatahi me te -4 i te rōpū tuarua.

$$x\left(x-20\right)-4\left(x-20\right)$$

Whakatauwehea atu te kīanga pātahi x-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-20\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x-4=0.

$$x=20 x=4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 24x mai i ngā taha e rua.

$$x^{2}+80-24x=0$$

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.

$$x^{2}-24x+80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -24 mō b, me 80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 80}}{2}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 80}}{2}$$

Whakareatia -4 ki te 80.

$$x=\frac{-\left(-24\right)±\sqrt{576-320}}{2}$$

Tāpiri 576 ki te -320.

$$x=\frac{-\left(-24\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-24\right)±16}{2}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±16}{2}$$

Nā, me whakaoti te whārite x=\frac{24±16}{2} ina he tāpiri te ±. Tāpiri 24 ki te 16.

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

Whakawehe 40 ki te 2.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{24±16}{2} ina he tango te ±. Tango 16 mai i 24.

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=20 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 24x mai i ngā taha e rua.

$$x^{2}+80-24x=0$$

Tangohia te 80 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}-24x=-80$$

Whakawehea te -24, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -12. Nā, tāpiria te pūrua o te -12 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-24x+\left(-12\right)^{2}=-80+\left(-12\right)^{2}$$

Pūrua -12.

$$x^{2}-24x+144=-80+144$$

Tāpiri -80 ki te 144.

$$x^{2}-24x+144=64$$

Tauwehea x^{2}-24x+144. 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}.

$$\left(x-12\right)^{2}=64$$

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

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x-12=8 x-12=-8$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$x=20 x=4$$