Lutasin ang x-1/x^2+4x+3+2/x^2+5x+6

$$\frac { x - 1 } { x ^ { 2 } + 4 x + 3 } + \frac { 2 } { x ^ { 2 } + 5 x + 6 }$$

I-evaluate

$$\frac{x}{\left(x+1\right)\left(x+2\right)}$$

Maiikling Hakbang sa Solution

I-factor out ang x^{2}+4x+3. I-factor out ang x^{2}+5x+6.

$$\frac{x-1}{\left(x+1\right)\left(x+3\right)}+\frac{2}{\left(x+2\right)\left(x+3\right)}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng \left(x+1\right)\left(x+3\right) at \left(x+2\right)\left(x+3\right) ay \left(x+1\right)\left(x+2\right)\left(x+3\right). I-multiply ang \frac{x-1}{\left(x+1\right)\left(x+3\right)} times \frac{x+2}{x+2}. I-multiply ang \frac{2}{\left(x+2\right)\left(x+3\right)} times \frac{x+1}{x+1}.

$$\frac{\left(x-1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}+\frac{2\left(x+1\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Dahil may parehong denominator ang \frac{\left(x-1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)} at \frac{2\left(x+1\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

$$\frac{\left(x-1\right)\left(x+2\right)+2\left(x+1\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Gawin ang mga pag-multiply sa \left(x-1\right)\left(x+2\right)+2\left(x+1\right).

$$\frac{x^{2}+2x-x-2+2x+2}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Pagsamahin ang magkakatulad na term sa x^{2}+2x-x-2+2x+2.

$$\frac{x^{2}+3x}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

I-factor ang mga expression na hindi pa na-factor sa \frac{x^{2}+3x}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}.

$$\frac{x\left(x+3\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

I-cancel out ang x+3 sa parehong numerator at denominator.

$$\frac{x}{\left(x+1\right)\left(x+2\right)}$$

Palawakin ang \left(x+1\right)\left(x+2\right).

$$\frac{x}{x^{2}+3x+2}$$

Palawakin

$$\frac{x}{\left(x+1\right)\left(x+2\right)}$$

Maiikling Hakbang sa Solution

I-factor out ang x^{2}+4x+3. I-factor out ang x^{2}+5x+6.

$$\frac{x-1}{\left(x+1\right)\left(x+3\right)}+\frac{2}{\left(x+2\right)\left(x+3\right)}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng \left(x+1\right)\left(x+3\right) at \left(x+2\right)\left(x+3\right) ay \left(x+1\right)\left(x+2\right)\left(x+3\right). I-multiply ang \frac{x-1}{\left(x+1\right)\left(x+3\right)} times \frac{x+2}{x+2}. I-multiply ang \frac{2}{\left(x+2\right)\left(x+3\right)} times \frac{x+1}{x+1}.

$$\frac{\left(x-1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}+\frac{2\left(x+1\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Dahil may parehong denominator ang \frac{\left(x-1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)} at \frac{2\left(x+1\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

$$\frac{\left(x-1\right)\left(x+2\right)+2\left(x+1\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Gawin ang mga pag-multiply sa \left(x-1\right)\left(x+2\right)+2\left(x+1\right).

$$\frac{x^{2}+2x-x-2+2x+2}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Pagsamahin ang magkakatulad na term sa x^{2}+2x-x-2+2x+2.

$$\frac{x^{2}+3x}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

I-factor ang mga expression na hindi pa na-factor sa \frac{x^{2}+3x}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}.

$$\frac{x\left(x+3\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

I-cancel out ang x+3 sa parehong numerator at denominator.

$$\frac{x}{\left(x+1\right)\left(x+2\right)}$$

Palawakin ang \left(x+1\right)\left(x+2\right).

$$\frac{x}{x^{2}+3x+2}$$