Solve x/x^2+3x+2-3=x-3/x+1

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

Solve for x

$$x = -\frac{7}{4} = -1\frac{3}{4} = -1.75$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+2\right), the least common multiple of x^{2}+3x+2,x+1.

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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

Use the distributive property to multiply x^{2}+3x+2 by -3.

$$x-3x^{2}-9x-6=\left(x+2\right)\left(x-3\right)$$

Combine x and -9x to get -8x.

$$-8x-3x^{2}-6=\left(x+2\right)\left(x-3\right)$$

Use the distributive property to multiply x+2 by x-3 and combine like terms.

$$-8x-3x^{2}-6=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-8x-3x^{2}-6-x^{2}=-x-6$$

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$-8x-4x^{2}-6=-x-6$$

Add x to both sides.

$$-8x-4x^{2}-6+x=-6$$

Combine -8x and x to get -7x.

$$-7x-4x^{2}-6=-6$$

Add 6 to both sides.

$$-7x-4x^{2}-6+6=0$$

Add -6 and 6 to get 0.

$$-7x-4x^{2}=0$$

Factor out x.

$$x\left(-7-4x\right)=0$$

To find equation solutions, solve x=0 and -7-4x=0.

$$x=0 x=-\frac{7}{4}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+2\right), the least common multiple of x^{2}+3x+2,x+1.

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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

Use the distributive property to multiply x^{2}+3x+2 by -3.

$$x-3x^{2}-9x-6=\left(x+2\right)\left(x-3\right)$$

Combine x and -9x to get -8x.

$$-8x-3x^{2}-6=\left(x+2\right)\left(x-3\right)$$

Use the distributive property to multiply x+2 by x-3 and combine like terms.

$$-8x-3x^{2}-6=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-8x-3x^{2}-6-x^{2}=-x-6$$

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$-8x-4x^{2}-6=-x-6$$

Add x to both sides.

$$-8x-4x^{2}-6+x=-6$$

Combine -8x and x to get -7x.

$$-7x-4x^{2}-6=-6$$

Add 6 to both sides.

$$-7x-4x^{2}-6+6=0$$

Add -6 and 6 to get 0.

$$-7x-4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4x^{2}-7x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}}}{2\left(-4\right)}$$

Take the square root of \left(-7\right)^{2}.

$$x=\frac{-\left(-7\right)±7}{2\left(-4\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±7}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{7±7}{-8}$$

Now solve the equation x=\frac{7±7}{-8} when ± is plus. Add 7 to 7.

$$x=\frac{14}{-8}$$

Reduce the fraction \frac{14}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{4}$$

Now solve the equation x=\frac{7±7}{-8} when ± is minus. Subtract 7 from 7.

$$x=\frac{0}{-8}$$

Divide 0 by -8.

$$x=0$$

The equation is now solved.

$$x=-\frac{7}{4} x=0$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+2\right), the least common multiple of x^{2}+3x+2,x+1.

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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

Use the distributive property to multiply x^{2}+3x+2 by -3.

$$x-3x^{2}-9x-6=\left(x+2\right)\left(x-3\right)$$

Combine x and -9x to get -8x.

$$-8x-3x^{2}-6=\left(x+2\right)\left(x-3\right)$$

Use the distributive property to multiply x+2 by x-3 and combine like terms.

$$-8x-3x^{2}-6=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-8x-3x^{2}-6-x^{2}=-x-6$$

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$-8x-4x^{2}-6=-x-6$$

Add x to both sides.

$$-8x-4x^{2}-6+x=-6$$

Combine -8x and x to get -7x.

$$-7x-4x^{2}-6=-6$$

Add 6 to both sides.

$$-7x-4x^{2}=-6+6$$

Add -6 and 6 to get 0.

$$-7x-4x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-4x^{2}-7x=0$$

Divide both sides by -4.

$$\frac{-4x^{2}-7x}{-4}=\frac{0}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{7}{-4}\right)x=\frac{0}{-4}$$

Divide -7 by -4.

$$x^{2}+\frac{7}{4}x=\frac{0}{-4}$$

Divide 0 by -4.

$$x^{2}+\frac{7}{4}x=0$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{49}{64}$$

Factor x^{2}+\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{49}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Simplify.

$$x+\frac{7}{8}=\frac{7}{8} x+\frac{7}{8}=-\frac{7}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$x=0 x=-\frac{7}{4}$$