Whakaoti i te x^2+b/ax+(b/2a)^2=-c/a+(b/2a)^2

$$x ^ { 2 } + \frac { b } { a } x + ( \frac { b } { 2 a } ) ^ { 2 } = - \frac { c } { a } + ( \frac { b } { 2 a } ) ^ { 2 }$$

Whakaoti mō b (complex solution)

$$\left\{\begin{matrix}b=-ax-\frac{c}{x}\text{, }&x\neq 0\text{ and }a\neq 0\\b\in \mathrm{C}\text{, }&c=0\text{ and }x=0\text{ and }a\neq 0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te a.

$$ax^{2}+bx+a\times \left(\frac{b}{2a}\right)^{2}=-c+a\times \left(\frac{b}{2a}\right)^{2}$$

Kia whakarewa i te \frac{b}{2a} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$ax^{2}+bx+a\times \frac{b^{2}}{\left(2a\right)^{2}}=-c+a\times \left(\frac{b}{2a}\right)^{2}$$

Tuhia te a\times \frac{b^{2}}{\left(2a\right)^{2}} hei hautanga kotahi.

$$ax^{2}+bx+\frac{ab^{2}}{\left(2a\right)^{2}}=-c+a\times \left(\frac{b}{2a}\right)^{2}$$

Kia whakarewa i te \frac{b}{2a} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$ax^{2}+bx+\frac{ab^{2}}{\left(2a\right)^{2}}=-c+a\times \frac{b^{2}}{\left(2a\right)^{2}}$$

Tuhia te a\times \frac{b^{2}}{\left(2a\right)^{2}} hei hautanga kotahi.

$$ax^{2}+bx+\frac{ab^{2}}{\left(2a\right)^{2}}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

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

$$ax^{2}+bx+\frac{ab^{2}}{2^{2}a^{2}}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

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

$$ax^{2}+bx+\frac{ab^{2}}{4a^{2}}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

Me whakakore tahi te a i te taurunga me te tauraro.

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

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

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{ab^{2}}{2^{2}a^{2}}$$

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

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{ab^{2}}{4a^{2}}$$

Me whakakore tahi te a i te taurunga me te tauraro.

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{b^{2}}{4a}$$

Tangohia te \frac{b^{2}}{4a} mai i ngā taha e rua.

$$ax^{2}+bx+\frac{b^{2}}{4a}-\frac{b^{2}}{4a}=-c$$

Whakareatia ngā taha e rua o te whārite ki te 4a.

$$ax^{2}\times 4a+bx\times 4a+b^{2}-b^{2}=-4ac$$

Whakaraupapatia anō ngā kīanga tau.

$$4aax^{2}+4abx+b^{2}-b^{2}=-4ac$$

Whakareatia te a ki te a, ka a^{2}.

$$4a^{2}x^{2}+4abx+b^{2}-b^{2}=-4ac$$

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

$$4a^{2}x^{2}+4abx=-4ac$$

Tangohia te 4a^{2}x^{2} mai i ngā taha e rua.

$$4abx=-4ac-4a^{2}x^{2}$$

He hanga arowhānui tō te whārite.

$$4axb=-4a^{2}x^{2}-4ac$$

Whakawehea ngā taha e rua ki te 4ax.

$$\frac{4axb}{4ax}=-\frac{4a\left(ax^{2}+c\right)}{4ax}$$

Mā te whakawehe ki te 4ax ka wetekia te whakareanga ki te 4ax.

$$b=-\frac{4a\left(ax^{2}+c\right)}{4ax}$$

Whakawehe -4a\left(c+ax^{2}\right) ki te 4ax.

$$b=-ax-\frac{c}{x}$$

Whakaoti mō b

$$\left\{\begin{matrix}b=-ax-\frac{c}{x}\text{, }&x\neq 0\text{ and }a\neq 0\\b\in \mathrm{R}\text{, }&c=0\text{ and }x=0\text{ and }a\neq 0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te a.

$$ax^{2}+bx+a\times \left(\frac{b}{2a}\right)^{2}=-c+a\times \left(\frac{b}{2a}\right)^{2}$$

Kia whakarewa i te \frac{b}{2a} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$ax^{2}+bx+a\times \frac{b^{2}}{\left(2a\right)^{2}}=-c+a\times \left(\frac{b}{2a}\right)^{2}$$

Tuhia te a\times \frac{b^{2}}{\left(2a\right)^{2}} hei hautanga kotahi.

$$ax^{2}+bx+\frac{ab^{2}}{\left(2a\right)^{2}}=-c+a\times \left(\frac{b}{2a}\right)^{2}$$

Kia whakarewa i te \frac{b}{2a} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$ax^{2}+bx+\frac{ab^{2}}{\left(2a\right)^{2}}=-c+a\times \frac{b^{2}}{\left(2a\right)^{2}}$$

Tuhia te a\times \frac{b^{2}}{\left(2a\right)^{2}} hei hautanga kotahi.

$$ax^{2}+bx+\frac{ab^{2}}{\left(2a\right)^{2}}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

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

$$ax^{2}+bx+\frac{ab^{2}}{2^{2}a^{2}}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

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

$$ax^{2}+bx+\frac{ab^{2}}{4a^{2}}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

Me whakakore tahi te a i te taurunga me te tauraro.

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{ab^{2}}{\left(2a\right)^{2}}$$

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

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{ab^{2}}{2^{2}a^{2}}$$

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

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{ab^{2}}{4a^{2}}$$

Me whakakore tahi te a i te taurunga me te tauraro.

$$ax^{2}+bx+\frac{b^{2}}{4a}=-c+\frac{b^{2}}{4a}$$

Tangohia te \frac{b^{2}}{4a} mai i ngā taha e rua.

$$ax^{2}+bx+\frac{b^{2}}{4a}-\frac{b^{2}}{4a}=-c$$

Whakareatia ngā taha e rua o te whārite ki te 4a.

$$ax^{2}\times 4a+bx\times 4a+b^{2}-b^{2}=-4ac$$

Whakaraupapatia anō ngā kīanga tau.

$$4aax^{2}+4abx+b^{2}-b^{2}=-4ac$$

Whakareatia te a ki te a, ka a^{2}.

$$4a^{2}x^{2}+4abx+b^{2}-b^{2}=-4ac$$

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

$$4a^{2}x^{2}+4abx=-4ac$$

Tangohia te 4a^{2}x^{2} mai i ngā taha e rua.

$$4abx=-4ac-4a^{2}x^{2}$$

He hanga arowhānui tō te whārite.

$$4axb=-4a^{2}x^{2}-4ac$$

Whakawehea ngā taha e rua ki te 4ax.

$$\frac{4axb}{4ax}=-\frac{4a\left(ax^{2}+c\right)}{4ax}$$

Mā te whakawehe ki te 4ax ka wetekia te whakareanga ki te 4ax.

$$b=-\frac{4a\left(ax^{2}+c\right)}{4ax}$$

Whakawehe -4a\left(c+ax^{2}\right) ki te 4ax.

$$b=-ax-\frac{c}{x}$$

Whakaoti mō a

$$\left\{\begin{matrix}a=-\frac{bx+c}{x^{2}}\text{, }&\left(c\neq 0\text{ or }b\neq 0\right)\text{ and }\left(b=0\text{ or }x\neq -\frac{c}{b}\right)\text{ and }x\neq 0\text{ and }c\neq -bx\\a\neq 0\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$