Whakaoti i te x^2=a(x-1)(x-2)+b(x-1)+c

$$x ^ { 2 } = a ( x - 1 ) ( x - 2 ) + b ( x - 1 ) + c$$

Whakaoti mō a (complex solution)

$$\left\{\begin{matrix}a=-\frac{-x^{2}+bx+c-b}{\left(x-2\right)\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\a\in \mathrm{C}\text{, }&\left(b=4-c\text{ and }x=2\right)\text{ or }\left(c=1\text{ and }x=1\right)\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te āhuatanga tohatoha hei whakarea te a ki te x-1.

$$x^{2}=\left(ax-a\right)\left(x-2\right)+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tuaritanga hei whakarea te ax-a ki te x-2 ka whakakotahi i ngā kupu rite.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tohatoha hei whakarea te b ki te x-1.

$$x^{2}=ax^{2}-3ax+2a+bx-b+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$ax^{2}-3ax+2a+bx-b+c=x^{2}$$

Tangohia te bx mai i ngā taha e rua.

$$ax^{2}-3ax+2a-b+c=x^{2}-bx$$

Me tāpiri te b ki ngā taha e rua.

$$ax^{2}-3ax+2a+c=x^{2}-bx+b$$

Tangohia te c mai i ngā taha e rua.

$$ax^{2}-3ax+2a=x^{2}-bx+b-c$$

Pahekotia ngā kīanga tau katoa e whai ana i te a.

$$\left(x^{2}-3x+2\right)a=x^{2}-bx+b-c$$

Whakawehea ngā taha e rua ki te x^{2}-3x+2.

$$\frac{\left(x^{2}-3x+2\right)a}{x^{2}-3x+2}=\frac{x^{2}-bx+b-c}{x^{2}-3x+2}$$

Mā te whakawehe ki te x^{2}-3x+2 ka wetekia te whakareanga ki te x^{2}-3x+2.

$$a=\frac{x^{2}-bx+b-c}{x^{2}-3x+2}$$

Whakawehe -bx+b+x^{2}-c ki te x^{2}-3x+2.

$$a=\frac{x^{2}-bx+b-c}{\left(x-2\right)\left(x-1\right)}$$

Whakaoti mō b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax^{2}-x^{2}-3ax+c+2a}{x-1}\text{, }&x\neq 1\\b\in \mathrm{C}\text{, }&c=1\text{ and }x=1\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te āhuatanga tohatoha hei whakarea te a ki te x-1.

$$x^{2}=\left(ax-a\right)\left(x-2\right)+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tuaritanga hei whakarea te ax-a ki te x-2 ka whakakotahi i ngā kupu rite.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tohatoha hei whakarea te b ki te x-1.

$$x^{2}=ax^{2}-3ax+2a+bx-b+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$ax^{2}-3ax+2a+bx-b+c=x^{2}$$

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

$$-3ax+2a+bx-b+c=x^{2}-ax^{2}$$

Me tāpiri te 3ax ki ngā taha e rua.

$$2a+bx-b+c=x^{2}-ax^{2}+3ax$$

Tangohia te 2a mai i ngā taha e rua.

$$bx-b+c=x^{2}-ax^{2}+3ax-2a$$

Tangohia te c mai i ngā taha e rua.

$$bx-b=x^{2}-ax^{2}+3ax-2a-c$$

Whakaraupapatia anō ngā kīanga tau.

$$bx-b=-ax^{2}+x^{2}+3ax-2a-c$$

Pahekotia ngā kīanga tau katoa e whai ana i te b.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-2a-c$$

He hanga arowhānui tō te whārite.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-c-2a$$

Whakawehea ngā taha e rua ki te x-1.

$$\frac{\left(x-1\right)b}{x-1}=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$

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

$$b=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$

Whakaoti mō a

$$\left\{\begin{matrix}a=-\frac{-x^{2}+bx+c-b}{\left(x-2\right)\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\a\in \mathrm{R}\text{, }&\left(b=4-c\text{ and }x=2\right)\text{ or }\left(c=1\text{ and }x=1\right)\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te āhuatanga tohatoha hei whakarea te a ki te x-1.

$$x^{2}=\left(ax-a\right)\left(x-2\right)+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tuaritanga hei whakarea te ax-a ki te x-2 ka whakakotahi i ngā kupu rite.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tohatoha hei whakarea te b ki te x-1.

$$x^{2}=ax^{2}-3ax+2a+bx-b+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$ax^{2}-3ax+2a+bx-b+c=x^{2}$$

Tangohia te bx mai i ngā taha e rua.

$$ax^{2}-3ax+2a-b+c=x^{2}-bx$$

Me tāpiri te b ki ngā taha e rua.

$$ax^{2}-3ax+2a+c=x^{2}-bx+b$$

Tangohia te c mai i ngā taha e rua.

$$ax^{2}-3ax+2a=x^{2}-bx+b-c$$

Pahekotia ngā kīanga tau katoa e whai ana i te a.

$$\left(x^{2}-3x+2\right)a=x^{2}-bx+b-c$$

Whakawehea ngā taha e rua ki te x^{2}-3x+2.

$$\frac{\left(x^{2}-3x+2\right)a}{x^{2}-3x+2}=\frac{x^{2}-bx+b-c}{x^{2}-3x+2}$$

Mā te whakawehe ki te x^{2}-3x+2 ka wetekia te whakareanga ki te x^{2}-3x+2.

$$a=\frac{x^{2}-bx+b-c}{x^{2}-3x+2}$$

Whakawehe x^{2}-bx+b-c ki te x^{2}-3x+2.

$$a=\frac{x^{2}-bx+b-c}{\left(x-2\right)\left(x-1\right)}$$

Whakaoti mō b

$$\left\{\begin{matrix}b=-\frac{ax^{2}-x^{2}-3ax+c+2a}{x-1}\text{, }&x\neq 1\\b\in \mathrm{R}\text{, }&c=1\text{ and }x=1\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te āhuatanga tohatoha hei whakarea te a ki te x-1.

$$x^{2}=\left(ax-a\right)\left(x-2\right)+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tuaritanga hei whakarea te ax-a ki te x-2 ka whakakotahi i ngā kupu rite.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

Whakamahia te āhuatanga tohatoha hei whakarea te b ki te x-1.

$$x^{2}=ax^{2}-3ax+2a+bx-b+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$ax^{2}-3ax+2a+bx-b+c=x^{2}$$

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

$$-3ax+2a+bx-b+c=x^{2}-ax^{2}$$

Me tāpiri te 3ax ki ngā taha e rua.

$$2a+bx-b+c=x^{2}-ax^{2}+3ax$$

Tangohia te 2a mai i ngā taha e rua.

$$bx-b+c=x^{2}-ax^{2}+3ax-2a$$

Tangohia te c mai i ngā taha e rua.

$$bx-b=x^{2}-ax^{2}+3ax-2a-c$$

Whakaraupapatia anō ngā kīanga tau.

$$bx-b=-ax^{2}+x^{2}+3ax-2a-c$$

Pahekotia ngā kīanga tau katoa e whai ana i te b.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-2a-c$$

He hanga arowhānui tō te whārite.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-c-2a$$

Whakawehea ngā taha e rua ki te x-1.

$$\frac{\left(x-1\right)b}{x-1}=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$

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

$$b=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$