Whakaoti i te x^2+49=14x

$$x ^ { 2 } + 49 = 14 x$$

Whakaoti mō x

$$x=7$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+49-14x=0$$

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

$$x^{2}-14x+49=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-14x+49 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.

$$a+b=-14 ab=49$$

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 49.

$$-1,-49 -7,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-49=-50 -7-7=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-7 b=-7$$

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

$$\left(x-7\right)\left(x-7\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x-7\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te x-7=0.

$$x=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+49-14x=0$$

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

$$x^{2}-14x+49=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+49. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=1\times 49=49$$

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 49.

$$-1,-49 -7,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-49=-50 -7-7=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-7 b=-7$$

Tuhia anō te x^{2}-14x+49 hei \left(x^{2}-7x\right)+\left(-7x+49\right).

$$\left(x^{2}-7x\right)+\left(-7x+49\right)$$

Tauwehea te x i te tuatahi me te -7 i te rōpū tuarua.

$$x\left(x-7\right)-7\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x-7\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x-7\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te x-7=0.

$$x=7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+49-14x=0$$

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.

$$x^{2}-14x+49=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me 49 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 49}}{2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 49}}{2}$$

Whakareatia -4 ki te 49.

$$x=\frac{-\left(-14\right)±\sqrt{196-196}}{2}$$

Tāpiri 196 ki te -196.

$$x=\frac{-\left(-14\right)±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-14}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14}{2}$$

Whakawehe 14 ki te 2.

$$x=7$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+49-14x=0$$

Tangohia te 49 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}-14x=-49$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-49+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-49+49$$

Tāpiri -49 ki te 49.

$$x^{2}-14x+49=0$$

Tauwehea te x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=0$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-7=0 x-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=7 x=7$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=7$$