Whakaoti i te v(v-10)-11(v-10)=0

$$v ( v - 10 ) - 11 ( v - 10 ) = 0$$

Whakaoti mō v

$$v=11$$

$$v=10$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te āhuatanga tohatoha hei whakarea te v ki te v-10.

$$v^{2}-10v-11\left(v-10\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -11 ki te v-10.

$$v^{2}-10v-11v+110=0$$

Pahekotia te -10v me -11v, ka -21v.

$$v^{2}-21v+110=0$$

Hei whakaoti i te whārite, whakatauwehea te v^{2}-21v+110 mā te whakamahi i te tātai v^{2}+\left(a+b\right)v+ab=\left(v+a\right)\left(v+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-21 ab=110$$

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 110.

$$-1,-110 -2,-55 -5,-22 -10,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-110=-111 -2-55=-57 -5-22=-27 -10-11=-21$$

Ko te otinga te takirua ka hoatu i te tapeke -21.

$$a=-11 b=-10$$

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

$$\left(v-11\right)\left(v-10\right)$$

Hei kimi otinga whārite, me whakaoti te v-11=0 me te v-10=0.

$$v=11 v=10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te v ki te v-10.

$$v^{2}-10v-11\left(v-10\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -11 ki te v-10.

$$v^{2}-10v-11v+110=0$$

Pahekotia te -10v me -11v, ka -21v.

$$v^{2}-21v+110=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei v^{2}+av+bv+110. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-21 ab=1\times 110=110$$

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 110.

$$-1,-110 -2,-55 -5,-22 -10,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-110=-111 -2-55=-57 -5-22=-27 -10-11=-21$$

Ko te otinga te takirua ka hoatu i te tapeke -21.

$$a=-11 b=-10$$

Tuhia anō te v^{2}-21v+110 hei \left(v^{2}-11v\right)+\left(-10v+110\right).

$$\left(v^{2}-11v\right)+\left(-10v+110\right)$$

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

$$v\left(v-11\right)-10\left(v-11\right)$$

Whakatauwehea atu te kīanga pātahi v-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(v-11\right)\left(v-10\right)$$

Hei kimi otinga whārite, me whakaoti te v-11=0 me te v-10=0.

$$v=11 v=10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te v ki te v-10.

$$v^{2}-10v-11\left(v-10\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -11 ki te v-10.

$$v^{2}-10v-11v+110=0$$

Pahekotia te -10v me -11v, ka -21v.

$$v^{2}-21v+110=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -21 mō b, me 110 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 110}}{2}$$

Pūrua -21.

$$v=\frac{-\left(-21\right)±\sqrt{441-4\times 110}}{2}$$

Whakareatia -4 ki te 110.

$$v=\frac{-\left(-21\right)±\sqrt{441-440}}{2}$$

Tāpiri 441 ki te -440.

$$v=\frac{-\left(-21\right)±\sqrt{1}}{2}$$

Tuhia te pūtakerua o te 1.

$$v=\frac{-\left(-21\right)±1}{2}$$

Ko te tauaro o -21 ko 21.

$$v=\frac{21±1}{2}$$

Nā, me whakaoti te whārite v=\frac{21±1}{2} ina he tāpiri te ±. Tāpiri 21 ki te 1.

$$v=\frac{22}{2}$$

Whakawehe 22 ki te 2.

$$v=11$$

Nā, me whakaoti te whārite v=\frac{21±1}{2} ina he tango te ±. Tango 1 mai i 21.

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

Whakawehe 20 ki te 2.

$$v=10$$

Kua oti te whārite te whakatau.

$$v=11 v=10$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te v ki te v-10.

$$v^{2}-10v-11\left(v-10\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -11 ki te v-10.

$$v^{2}-10v-11v+110=0$$

Pahekotia te -10v me -11v, ka -21v.

$$v^{2}-21v+110=0$$

Tangohia te 110 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$v^{2}-21v=-110$$

Whakawehea te -21, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{21}{2}. Nā, tāpiria te pūrua o te -\frac{21}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$v^{2}-21v+\left(-\frac{21}{2}\right)^{2}=-110+\left(-\frac{21}{2}\right)^{2}$$

Pūruatia -\frac{21}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$v^{2}-21v+\frac{441}{4}=-110+\frac{441}{4}$$

Tāpiri -110 ki te \frac{441}{4}.

$$v^{2}-21v+\frac{441}{4}=\frac{1}{4}$$

Tauwehea v^{2}-21v+\frac{441}{4}. 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}.

$$\left(v-\frac{21}{2}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(v-\frac{21}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$v-\frac{21}{2}=\frac{1}{2} v-\frac{21}{2}=-\frac{1}{2}$$

Me tāpiri \frac{21}{2} ki ngā taha e rua o te whārite.

$$v=11 v=10$$