Whakaoti i te x^2+45=14x

$$x ^ { 2 } + 45 = 14 x$$

Whakaoti mō x

$$x=5$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+45-14x=0$$

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

$$x^{2}-14x+45=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-14x+45 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.

$$a+b=-14 ab=45$$

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 45.

$$-1,-45 -3,-15 -5,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-45=-46 -3-15=-18 -5-9=-14$$

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

$$a=-9 b=-5$$

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

$$\left(x-9\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x-5=0.

$$x=9 x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+45-14x=0$$

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

$$x^{2}-14x+45=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=1\times 45=45$$

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 45.

$$-1,-45 -3,-15 -5,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-45=-46 -3-15=-18 -5-9=-14$$

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

$$a=-9 b=-5$$

Tuhia anō te x^{2}-14x+45 hei \left(x^{2}-9x\right)+\left(-5x+45\right).

$$\left(x^{2}-9x\right)+\left(-5x+45\right)$$

Tauwehea te x i te tuatahi me te -5 i te rōpū tuarua.

$$x\left(x-9\right)-5\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x-5=0.

$$x=9 x=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+45-14x=0$$

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.

$$x^{2}-14x+45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me 45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 45}}{2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 45}}{2}$$

Whakareatia -4 ki te 45.

$$x=\frac{-\left(-14\right)±\sqrt{196-180}}{2}$$

Tāpiri 196 ki te -180.

$$x=\frac{-\left(-14\right)±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-\left(-14\right)±4}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±4}{2}$$

Nā, me whakaoti te whārite x=\frac{14±4}{2} ina he tāpiri te ±. Tāpiri 14 ki te 4.

$$x=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{14±4}{2} ina he tango te ±. Tango 4 mai i 14.

$$x=\frac{10}{2}$$

Whakawehe 10 ki te 2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=9 x=5$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 14x mai i ngā taha e rua.

$$x^{2}+45-14x=0$$

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

$$x^{2}-14x=-45$$

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.

$$x^{2}-14x+\left(-7\right)^{2}=-45+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-45+49$$

Tāpiri -45 ki te 49.

$$x^{2}-14x+49=4$$

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

$$\left(x-7\right)^{2}=4$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-7=2 x-7=-2$$

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

$$x=9 x=5$$