Whakaoti i te [(2m-n)^2+(m-2n)(m+2n)-5m(m+n)]=(-3n)

$$[ ( 2 m - n ) ^ { 2 } + ( m - 2 n ) ( m + 2 n ) - 5 m ( m + n ) ] = ( - 3 n )$$

Whakaoti mō m (complex solution)

$$\left\{\begin{matrix}\\m=\frac{1-n}{3}\text{, }&\text{unconditionally}\\m\in \mathrm{C}\text{, }&n=0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2m-n\right)^{2}.

$$4m^{2}-4mn+n^{2}+\left(m-2n\right)\left(m+2n\right)-5m\left(m+n\right)=-3n$$

Whakaarohia te \left(m-2n\right)\left(m+2n\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4m^{2}-4mn+n^{2}+m^{2}-\left(2n\right)^{2}-5m\left(m+n\right)=-3n$$

Whakarohaina te \left(2n\right)^{2}.

$$4m^{2}-4mn+n^{2}+m^{2}-2^{2}n^{2}-5m\left(m+n\right)=-3n$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4m^{2}-4mn+n^{2}+m^{2}-4n^{2}-5m\left(m+n\right)=-3n$$

Pahekotia te 4m^{2} me m^{2}, ka 5m^{2}.

$$5m^{2}-4mn+n^{2}-4n^{2}-5m\left(m+n\right)=-3n$$

Pahekotia te n^{2} me -4n^{2}, ka -3n^{2}.

$$5m^{2}-4mn-3n^{2}-5m\left(m+n\right)=-3n$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5m ki te m+n.

$$5m^{2}-4mn-3n^{2}-5m^{2}-5mn=-3n$$

Pahekotia te 5m^{2} me -5m^{2}, ka 0.

$$-4mn-3n^{2}-5mn=-3n$$

Pahekotia te -4mn me -5mn, ka -9mn.

$$-9mn-3n^{2}=-3n$$

Me tāpiri te 3n^{2} ki ngā taha e rua.

$$-9mn=-3n+3n^{2}$$

He hanga arowhānui tō te whārite.

$$\left(-9n\right)m=3n^{2}-3n$$

Whakawehea ngā taha e rua ki te -9n.

$$\frac{\left(-9n\right)m}{-9n}=\frac{3n\left(n-1\right)}{-9n}$$

Mā te whakawehe ki te -9n ka wetekia te whakareanga ki te -9n.

$$m=\frac{3n\left(n-1\right)}{-9n}$$

Whakawehe 3n\left(-1+n\right) ki te -9n.

$$m=\frac{1-n}{3}$$

Whakaoti mō m

$$\left\{\begin{matrix}\\m=\frac{1-n}{3}\text{, }&\text{unconditionally}\\m\in \mathrm{R}\text{, }&n=0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2m-n\right)^{2}.

$$4m^{2}-4mn+n^{2}+\left(m-2n\right)\left(m+2n\right)-5m\left(m+n\right)=-3n$$

Whakaarohia te \left(m-2n\right)\left(m+2n\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4m^{2}-4mn+n^{2}+m^{2}-\left(2n\right)^{2}-5m\left(m+n\right)=-3n$$

Whakarohaina te \left(2n\right)^{2}.

$$4m^{2}-4mn+n^{2}+m^{2}-2^{2}n^{2}-5m\left(m+n\right)=-3n$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4m^{2}-4mn+n^{2}+m^{2}-4n^{2}-5m\left(m+n\right)=-3n$$

Pahekotia te 4m^{2} me m^{2}, ka 5m^{2}.

$$5m^{2}-4mn+n^{2}-4n^{2}-5m\left(m+n\right)=-3n$$

Pahekotia te n^{2} me -4n^{2}, ka -3n^{2}.

$$5m^{2}-4mn-3n^{2}-5m\left(m+n\right)=-3n$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5m ki te m+n.

$$5m^{2}-4mn-3n^{2}-5m^{2}-5mn=-3n$$

Pahekotia te 5m^{2} me -5m^{2}, ka 0.

$$-4mn-3n^{2}-5mn=-3n$$

Pahekotia te -4mn me -5mn, ka -9mn.

$$-9mn-3n^{2}=-3n$$

Me tāpiri te 3n^{2} ki ngā taha e rua.

$$-9mn=-3n+3n^{2}$$

He hanga arowhānui tō te whārite.

$$\left(-9n\right)m=3n^{2}-3n$$

Whakawehea ngā taha e rua ki te -9n.

$$\frac{\left(-9n\right)m}{-9n}=\frac{3n\left(n-1\right)}{-9n}$$

Mā te whakawehe ki te -9n ka wetekia te whakareanga ki te -9n.

$$m=\frac{3n\left(n-1\right)}{-9n}$$

Whakawehe 3n\left(-1+n\right) ki te -9n.

$$m=\frac{1-n}{3}$$

Whakaoti mō n

$$n=1-3m$$

$$n=0$$