Solve 3-x/left(x+1right)left(x+2right)=15/1

$$\frac{ 3-x }{ \left( x+1 \right) \left( x+2 \right) } = \frac{ 15 }{ 1 }$$

Solve for x

$$x=\frac{2\sqrt{31}-23}{15}\approx -0.790964752$$

$$x=\frac{-2\sqrt{31}-23}{15}\approx -2.275701915$$

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).

$$3-x=\left(x+1\right)\left(x+2\right)\times 15$$

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

$$3-x=\left(x^{2}+3x+2\right)\times 15$$

Use the distributive property to multiply x^{2}+3x+2 by 15.

$$3-x=15x^{2}+45x+30$$

Subtract 15x^{2} from both sides.

$$3-x-15x^{2}=45x+30$$

Subtract 45x from both sides.

$$3-x-15x^{2}-45x=30$$

Combine -x and -45x to get -46x.

$$3-46x-15x^{2}=30$$

Subtract 30 from both sides.

$$3-46x-15x^{2}-30=0$$

Subtract 30 from 3 to get -27.

$$-27-46x-15x^{2}=0$$

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.

$$-15x^{2}-46x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, -46 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\left(-15\right)\left(-27\right)}}{2\left(-15\right)}$$

Square -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\left(-15\right)\left(-27\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-\left(-46\right)±\sqrt{2116+60\left(-27\right)}}{2\left(-15\right)}$$

Multiply 60 times -27.

$$x=\frac{-\left(-46\right)±\sqrt{2116-1620}}{2\left(-15\right)}$$

Add 2116 to -1620.

$$x=\frac{-\left(-46\right)±\sqrt{496}}{2\left(-15\right)}$$

Take the square root of 496.

$$x=\frac{-\left(-46\right)±4\sqrt{31}}{2\left(-15\right)}$$

The opposite of -46 is 46.

$$x=\frac{46±4\sqrt{31}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{46±4\sqrt{31}}{-30}$$

Now solve the equation x=\frac{46±4\sqrt{31}}{-30} when ± is plus. Add 46 to 4\sqrt{31}.

$$x=\frac{4\sqrt{31}+46}{-30}$$

Divide 46+4\sqrt{31} by -30.

$$x=\frac{-2\sqrt{31}-23}{15}$$

Now solve the equation x=\frac{46±4\sqrt{31}}{-30} when ± is minus. Subtract 4\sqrt{31} from 46.

$$x=\frac{46-4\sqrt{31}}{-30}$$

Divide 46-4\sqrt{31} by -30.

$$x=\frac{2\sqrt{31}-23}{15}$$

The equation is now solved.

$$x=\frac{-2\sqrt{31}-23}{15} x=\frac{2\sqrt{31}-23}{15}$$

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).

$$3-x=\left(x+1\right)\left(x+2\right)\times 15$$

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

$$3-x=\left(x^{2}+3x+2\right)\times 15$$

Use the distributive property to multiply x^{2}+3x+2 by 15.

$$3-x=15x^{2}+45x+30$$

Subtract 15x^{2} from both sides.

$$3-x-15x^{2}=45x+30$$

Subtract 45x from both sides.

$$3-x-15x^{2}-45x=30$$

Combine -x and -45x to get -46x.

$$3-46x-15x^{2}=30$$

Subtract 3 from both sides.

$$-46x-15x^{2}=30-3$$

Subtract 3 from 30 to get 27.

$$-46x-15x^{2}=27$$

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.

$$-15x^{2}-46x=27$$

Divide both sides by -15.

$$\frac{-15x^{2}-46x}{-15}=\frac{27}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\left(-\frac{46}{-15}\right)x=\frac{27}{-15}$$

Divide -46 by -15.

$$x^{2}+\frac{46}{15}x=\frac{27}{-15}$$

Reduce the fraction \frac{27}{-15} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{46}{15}x=-\frac{9}{5}$$

Divide \frac{46}{15}, the coefficient of the x term, by 2 to get \frac{23}{15}. Then add the square of \frac{23}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{46}{15}x+\left(\frac{23}{15}\right)^{2}=-\frac{9}{5}+\left(\frac{23}{15}\right)^{2}$$

Square \frac{23}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{46}{15}x+\frac{529}{225}=-\frac{9}{5}+\frac{529}{225}$$

Add -\frac{9}{5} to \frac{529}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{46}{15}x+\frac{529}{225}=\frac{124}{225}$$

Factor x^{2}+\frac{46}{15}x+\frac{529}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{23}{15}\right)^{2}=\frac{124}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23}{15}\right)^{2}}=\sqrt{\frac{124}{225}}$$

Simplify.

$$x+\frac{23}{15}=\frac{2\sqrt{31}}{15} x+\frac{23}{15}=-\frac{2\sqrt{31}}{15}$$

Subtract \frac{23}{15} from both sides of the equation.

$$x=\frac{2\sqrt{31}-23}{15} x=\frac{-2\sqrt{31}-23}{15}$$