Whakaoti i te x^2-64x+348=0

$${ x }^{ 2 } -64x+348=0$$

Whakaoti mō x

$$x=6$$

$$x=58$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-64x+348 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.

$$a+b=-64 ab=348$$

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 348.

$$-1,-348 -2,-174 -3,-116 -4,-87 -6,-58 -12,-29$$

Tātaihia te tapeke mō ia takirua.

$$-1-348=-349 -2-174=-176 -3-116=-119 -4-87=-91 -6-58=-64 -12-29=-41$$

Ko te otinga te takirua ka hoatu i te tapeke -64.

$$a=-58 b=-6$$

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

$$\left(x-58\right)\left(x-6\right)$$

Hei kimi otinga whārite, me whakaoti te x-58=0 me te x-6=0.

$$x=58 x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+348. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-64 ab=1\times 348=348$$

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 348.

$$-1,-348 -2,-174 -3,-116 -4,-87 -6,-58 -12,-29$$

Tātaihia te tapeke mō ia takirua.

$$-1-348=-349 -2-174=-176 -3-116=-119 -4-87=-91 -6-58=-64 -12-29=-41$$

Ko te otinga te takirua ka hoatu i te tapeke -64.

$$a=-58 b=-6$$

Tuhia anō te x^{2}-64x+348 hei \left(x^{2}-58x\right)+\left(-6x+348\right).

$$\left(x^{2}-58x\right)+\left(-6x+348\right)$$

Tauwehea te x i te tuatahi me te -6 i te rōpū tuarua.

$$x\left(x-58\right)-6\left(x-58\right)$$

Whakatauwehea atu te kīanga pātahi x-58 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-58\right)\left(x-6\right)$$

Hei kimi otinga whārite, me whakaoti te x-58=0 me te x-6=0.

$$x=58 x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}-64x+348=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -64 mō b, me 348 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 348}}{2}$$

Pūrua -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 348}}{2}$$

Whakareatia -4 ki te 348.

$$x=\frac{-\left(-64\right)±\sqrt{4096-1392}}{2}$$

Tāpiri 4096 ki te -1392.

$$x=\frac{-\left(-64\right)±\sqrt{2704}}{2}$$

Tuhia te pūtakerua o te 2704.

$$x=\frac{-\left(-64\right)±52}{2}$$

Ko te tauaro o -64 ko 64.

$$x=\frac{64±52}{2}$$

Nā, me whakaoti te whārite x=\frac{64±52}{2} ina he tāpiri te ±. Tāpiri 64 ki te 52.

$$x=\frac{116}{2}$$

Whakawehe 116 ki te 2.

$$x=58$$

Nā, me whakaoti te whārite x=\frac{64±52}{2} ina he tango te ±. Tango 52 mai i 64.

$$x=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=58 x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$x^{2}-64x+348=0$$

Me tango 348 mai i ngā taha e rua o te whārite.

$$x^{2}-64x+348-348=-348$$

Mā te tango i te 348 i a ia ake anō ka toe ko te 0.

$$x^{2}-64x=-348$$

Whakawehea te -64, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -32. Nā, tāpiria te pūrua o te -32 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-64x+\left(-32\right)^{2}=-348+\left(-32\right)^{2}$$

Pūrua -32.

$$x^{2}-64x+1024=-348+1024$$

Tāpiri -348 ki te 1024.

$$x^{2}-64x+1024=676$$

Tauwehea x^{2}-64x+1024. 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}.

$$\left(x-32\right)^{2}=676$$

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

$$\sqrt{\left(x-32\right)^{2}}=\sqrt{676}$$

Whakarūnātia.

$$x-32=26 x-32=-26$$

Me tāpiri 32 ki ngā taha e rua o te whārite.

$$x=58 x=6$$