Whakaoti i te 12(n-4)-30=n^2-9n+12

$$12 ( n - 4 ) - 30 = n ^ { 2 } - 9 n + 12$$

Whakaoti mō n

$$n=6$$

$$n=15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te n-4.

$$12n-48-30=n^{2}-9n+12$$

Tangohia te 30 i te -48, ka -78.

$$12n-78=n^{2}-9n+12$$

Tangohia te n^{2} mai i ngā taha e rua.

$$12n-78-n^{2}=-9n+12$$

Me tāpiri te 9n ki ngā taha e rua.

$$12n-78-n^{2}+9n=12$$

Pahekotia te 12n me 9n, ka 21n.

$$21n-78-n^{2}=12$$

Tangohia te 12 mai i ngā taha e rua.

$$21n-78-n^{2}-12=0$$

Tangohia te 12 i te -78, ka -90.

$$21n-90-n^{2}=0$$

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

$$-n^{2}+21n-90=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -n^{2}+an+bn-90. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=21 ab=-\left(-90\right)=90$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 90.

$$1,90 2,45 3,30 5,18 6,15 9,10$$

Tātaihia te tapeke mō ia takirua.

$$1+90=91 2+45=47 3+30=33 5+18=23 6+15=21 9+10=19$$

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

$$a=15 b=6$$

Tuhia anō te -n^{2}+21n-90 hei \left(-n^{2}+15n\right)+\left(6n-90\right).

$$\left(-n^{2}+15n\right)+\left(6n-90\right)$$

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

$$-n\left(n-15\right)+6\left(n-15\right)$$

Whakatauwehea atu te kīanga pātahi n-15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-15\right)\left(-n+6\right)$$

Hei kimi otinga whārite, me whakaoti te n-15=0 me te -n+6=0.

$$n=15 n=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te n-4.

$$12n-48-30=n^{2}-9n+12$$

Tangohia te 30 i te -48, ka -78.

$$12n-78=n^{2}-9n+12$$

Tangohia te n^{2} mai i ngā taha e rua.

$$12n-78-n^{2}=-9n+12$$

Me tāpiri te 9n ki ngā taha e rua.

$$12n-78-n^{2}+9n=12$$

Pahekotia te 12n me 9n, ka 21n.

$$21n-78-n^{2}=12$$

Tangohia te 12 mai i ngā taha e rua.

$$21n-78-n^{2}-12=0$$

Tangohia te 12 i te -78, ka -90.

$$21n-90-n^{2}=0$$

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.

$$-n^{2}+21n-90=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 21 mō b, me -90 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-21±\sqrt{21^{2}-4\left(-1\right)\left(-90\right)}}{2\left(-1\right)}$$

Pūrua 21.

$$n=\frac{-21±\sqrt{441-4\left(-1\right)\left(-90\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$n=\frac{-21±\sqrt{441+4\left(-90\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -90.

$$n=\frac{-21±\sqrt{441-360}}{2\left(-1\right)}$$

Tāpiri 441 ki te -360.

$$n=\frac{-21±\sqrt{81}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 81.

$$n=\frac{-21±9}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$n=\frac{-21±9}{-2}$$

Nā, me whakaoti te whārite n=\frac{-21±9}{-2} ina he tāpiri te ±. Tāpiri -21 ki te 9.

$$n=-\frac{12}{-2}$$

Whakawehe -12 ki te -2.

$$n=6$$

Nā, me whakaoti te whārite n=\frac{-21±9}{-2} ina he tango te ±. Tango 9 mai i -21.

$$n=-\frac{30}{-2}$$

Whakawehe -30 ki te -2.

$$n=15$$

Kua oti te whārite te whakatau.

$$n=6 n=15$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te n-4.

$$12n-48-30=n^{2}-9n+12$$

Tangohia te 30 i te -48, ka -78.

$$12n-78=n^{2}-9n+12$$

Tangohia te n^{2} mai i ngā taha e rua.

$$12n-78-n^{2}=-9n+12$$

Me tāpiri te 9n ki ngā taha e rua.

$$12n-78-n^{2}+9n=12$$

Pahekotia te 12n me 9n, ka 21n.

$$21n-78-n^{2}=12$$

Me tāpiri te 78 ki ngā taha e rua.

$$21n-n^{2}=12+78$$

Tāpirihia te 12 ki te 78, ka 90.

$$21n-n^{2}=90$$

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.

$$-n^{2}+21n=90$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-n^{2}+21n}{-1}=\frac{90}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$n^{2}+\frac{21}{-1}n=\frac{90}{-1}$$

Whakawehe 21 ki te -1.

$$n^{2}-21n=\frac{90}{-1}$$

Whakawehe 90 ki te -1.

$$n^{2}-21n=-90$$

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.

$$n^{2}-21n+\left(-\frac{21}{2}\right)^{2}=-90+\left(-\frac{21}{2}\right)^{2}$$

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

$$n^{2}-21n+\frac{441}{4}=-90+\frac{441}{4}$$

Tāpiri -90 ki te \frac{441}{4}.

$$n^{2}-21n+\frac{441}{4}=\frac{81}{4}$$

Tauwehea n^{2}-21n+\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}.

$$\left(n-\frac{21}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(n-\frac{21}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

$$n-\frac{21}{2}=\frac{9}{2} n-\frac{21}{2}=-\frac{9}{2}$$

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

$$n=15 n=6$$