Whakaoti i te 3/2x+1-1/3x+2=2g

$$\frac { 3 } { 2 x + 1 } - \frac { 1 } { 3 x + 2 } = 2 g$$

Whakaoti mō g

$$g=\frac{7x+5}{2\left(6x^{2}+7x+2\right)}$$

$$x\neq -\frac{1}{2}\text{ and }x\neq -\frac{2}{3}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Me whakarea ngā taha e rua o te whārite ki te \left(2x+1\right)\left(3x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2x+1,3x+2.

$$\left(3x+2\right)\times 3-\left(2x+1\right)=2g\left(2x+1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+2 ki te 3.

$$9x+6-\left(2x+1\right)=2g\left(2x+1\right)\left(3x+2\right)$$

Hei kimi i te tauaro o 2x+1, kimihia te tauaro o ia taurangi.

$$9x+6-2x-1=2g\left(2x+1\right)\left(3x+2\right)$$

Pahekotia te 9x me -2x, ka 7x.

$$7x+6-1=2g\left(2x+1\right)\left(3x+2\right)$$

Tangohia te 1 i te 6, ka 5.

$$7x+5=2g\left(2x+1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2g ki te 2x+1.

$$7x+5=\left(4xg+2g\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4xg+2g ki te 3x+2 ka whakakotahi i ngā kupu rite.

$$7x+5=12gx^{2}+14gx+4g$$

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

$$12gx^{2}+14gx+4g=7x+5$$

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

$$\left(12x^{2}+14x+4\right)g=7x+5$$

Whakawehea ngā taha e rua ki te 12x^{2}+14x+4.

$$\frac{\left(12x^{2}+14x+4\right)g}{12x^{2}+14x+4}=\frac{7x+5}{12x^{2}+14x+4}$$

Mā te whakawehe ki te 12x^{2}+14x+4 ka wetekia te whakareanga ki te 12x^{2}+14x+4.

$$g=\frac{7x+5}{12x^{2}+14x+4}$$

Whakawehe 7x+5 ki te 12x^{2}+14x+4.

$$g=\frac{7x+5}{2\left(6x^{2}+7x+2\right)}$$

Whakaoti mō x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{; }x=-\frac{-\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{, }&g\neq 0\\x=-\frac{5}{7}\text{, }&g=0\end{matrix}\right.$$

Whakaoti mō x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{; }x=-\frac{-\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{, }&g\leq -3\sqrt{2}-\frac{11}{2}\text{ or }\left(g\neq 0\text{ and }g\geq 3\sqrt{2}-\frac{11}{2}\right)\\x=-\frac{5}{7}\text{, }&g=0\end{matrix}\right.$$