Solve 3-x/left(x+1right)left(x+2right)=1.5/1

$$\frac{ 3-x }{ \left( x+1 \right) \left( x+2 \right) } = \frac{ 1.5 }{ 1 }$$

Solve for x

$$x = -\frac{11}{3} = -3\frac{2}{3} \approx -3.666666667$$

$$x=0$$

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).

$$3-x=\left(x+1\right)\left(x+2\right)\times 1.5$$

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

$$3-x=\left(x^{2}+3x+2\right)\times 1.5$$

Use the distributive property to multiply x^{2}+3x+2 by 1.5.

$$3-x=1.5x^{2}+4.5x+3$$

Subtract 1.5x^{2} from both sides.

$$3-x-1.5x^{2}=4.5x+3$$

Subtract 4.5x from both sides.

$$3-x-1.5x^{2}-4.5x=3$$

Combine -x and -4.5x to get -5.5x.

$$3-5.5x-1.5x^{2}=3$$

Subtract 3 from both sides.

$$3-5.5x-1.5x^{2}-3=0$$

Subtract 3 from 3 to get 0.

$$-5.5x-1.5x^{2}=0$$

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.

$$-1.5x^{2}-5.5x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1.5 for a, -5.5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5.5\right)±\sqrt{\left(-5.5\right)^{2}}}{2\left(-1.5\right)}$$

Take the square root of \left(-5.5\right)^{2}.

$$x=\frac{-\left(-5.5\right)±\frac{11}{2}}{2\left(-1.5\right)}$$

The opposite of -5.5 is 5.5.

$$x=\frac{5.5±\frac{11}{2}}{2\left(-1.5\right)}$$

Multiply 2 times -1.5.

$$x=\frac{5.5±\frac{11}{2}}{-3}$$

Now solve the equation x=\frac{5.5±\frac{11}{2}}{-3} when ± is plus. Add 5.5 to \frac{11}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{11}{-3}$$

Divide 11 by -3.

$$x=-\frac{11}{3}$$

Now solve the equation x=\frac{5.5±\frac{11}{2}}{-3} when ± is minus. Subtract \frac{11}{2} from 5.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-3}$$

Divide 0 by -3.

$$x=0$$

The equation is now solved.

$$x=-\frac{11}{3} x=0$$

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).

$$3-x=\left(x+1\right)\left(x+2\right)\times 1.5$$

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

$$3-x=\left(x^{2}+3x+2\right)\times 1.5$$

Use the distributive property to multiply x^{2}+3x+2 by 1.5.

$$3-x=1.5x^{2}+4.5x+3$$

Subtract 1.5x^{2} from both sides.

$$3-x-1.5x^{2}=4.5x+3$$

Subtract 4.5x from both sides.

$$3-x-1.5x^{2}-4.5x=3$$

Combine -x and -4.5x to get -5.5x.

$$3-5.5x-1.5x^{2}=3$$

Subtract 3 from both sides.

$$-5.5x-1.5x^{2}=3-3$$

Subtract 3 from 3 to get 0.

$$-5.5x-1.5x^{2}=0$$

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.

$$-1.5x^{2}-5.5x=0$$

Divide both sides of the equation by -1.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-1.5x^{2}-5.5x}{-1.5}=\frac{0}{-1.5}$$

Dividing by -1.5 undoes the multiplication by -1.5.

$$x^{2}+\left(-\frac{5.5}{-1.5}\right)x=\frac{0}{-1.5}$$

Divide -5.5 by -1.5 by multiplying -5.5 by the reciprocal of -1.5.

$$x^{2}+\frac{11}{3}x=\frac{0}{-1.5}$$

Divide 0 by -1.5 by multiplying 0 by the reciprocal of -1.5.

$$x^{2}+\frac{11}{3}x=0$$

Divide \frac{11}{3}, the coefficient of the x term, by 2 to get \frac{11}{6}. Then add the square of \frac{11}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{3}x+\frac{11}{6}^{2}=\frac{11}{6}^{2}$$

Square \frac{11}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=\frac{121}{36}$$

Factor x^{2}+\frac{11}{3}x+\frac{121}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{6}\right)^{2}=\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Simplify.

$$x+\frac{11}{6}=\frac{11}{6} x+\frac{11}{6}=-\frac{11}{6}$$

Subtract \frac{11}{6} from both sides of the equation.

$$x=0 x=-\frac{11}{3}$$