Whakaoti i te z^2-20z=-100

$$z ^ { 2 } - 20 z = - 100$$

Whakaoti mō z

$$z=10$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 100 ki ngā taha e rua.

$$z^{2}-20z+100=0$$

Hei whakaoti i te whārite, whakatauwehea te z^{2}-20z+100 mā te whakamahi i te tātai z^{2}+\left(a+b\right)z+ab=\left(z+a\right)\left(z+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-20 ab=100$$

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 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-10 b=-10$$

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

$$\left(z-10\right)\left(z-10\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(z-10\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te z-10=0.

$$z=10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 100 ki ngā taha e rua.

$$z^{2}-20z+100=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei z^{2}+az+bz+100. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-20 ab=1\times 100=100$$

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 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-10 b=-10$$

Tuhia anō te z^{2}-20z+100 hei \left(z^{2}-10z\right)+\left(-10z+100\right).

$$\left(z^{2}-10z\right)+\left(-10z+100\right)$$

Tauwehea te z i te tuatahi me te -10 i te rōpū tuarua.

$$z\left(z-10\right)-10\left(z-10\right)$$

Whakatauwehea atu te kīanga pātahi z-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(z-10\right)\left(z-10\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(z-10\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te z-10=0.

$$z=10$$

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.

$$z^{2}-20z=-100$$

Me tāpiri 100 ki ngā taha e rua o te whārite.

$$z^{2}-20z-\left(-100\right)=-100-\left(-100\right)$$

Mā te tango i te -100 i a ia ake anō ka toe ko te 0.

$$z^{2}-20z-\left(-100\right)=0$$

Tango -100 mai i 0.

$$z^{2}-20z+100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -20 mō b, me 100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 100}}{2}$$

Pūrua -20.

$$z=\frac{-\left(-20\right)±\sqrt{400-4\times 100}}{2}$$

Whakareatia -4 ki te 100.

$$z=\frac{-\left(-20\right)±\sqrt{400-400}}{2}$$

Tāpiri 400 ki te -400.

$$z=\frac{-\left(-20\right)±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

$$z=-\frac{-20}{2}$$

Ko te tauaro o -20 ko 20.

$$z=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$z=10$$

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.

$$z^{2}-20z=-100$$

Whakawehea te -20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -10. Nā, tāpiria te pūrua o te -10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}-20z+\left(-10\right)^{2}=-100+\left(-10\right)^{2}$$

Pūrua -10.

$$z^{2}-20z+100=-100+100$$

Tāpiri -100 ki te 100.

$$z^{2}-20z+100=0$$

Tauwehea z^{2}-20z+100. 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}.

$$\left(z-10\right)^{2}=0$$

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

$$\sqrt{\left(z-10\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$z-10=0 z-10=0$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$z=10 z=10$$

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

$$z=10$$