Solve 2x/-(x-2)-1=-x/x+3+5(1+2x)/6-x-x^2

$$\frac { 2 x } { - ( x - 2 ) } - 1 = \frac { - x } { x + 3 } + \frac { 5 ( 1 + 2 x ) } { 6 - x - x ^ { 2 } }$$

Solve for x

$$x = \frac{\sqrt{89} + 1}{4} \approx 2.608495283$$

$$x=\frac{1-\sqrt{89}}{4}\approx -2.108495283$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+3\right), the least common multiple of x+3,6-x-x^{2}.

$$-\left(x+3\right)\times 2x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

To find the opposite of x+3, find the opposite of each term.

$$\left(-x-3\right)\times 2x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply -x-3 by 2.

$$\left(-2x-6\right)x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply -2x-6 by x.

$$-2x^{2}-6x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

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

$$-2x^{2}-6x+\left(x^{2}+x-6\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply x^{2}+x-6 by -1.

$$-2x^{2}-6x-x^{2}-x+6=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

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

$$-3x^{2}-6x-x+6=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Combine -6x and -x to get -7x.

$$-3x^{2}-7x+6=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply x-2 by -x.

$$-3x^{2}-7x+6=x\left(-x\right)-2\left(-x\right)-5\left(1+2x\right)$$

Multiply -2 and -1 to get 2.

$$-3x^{2}-7x+6=x\left(-x\right)+2x-5\left(1+2x\right)$$

Use the distributive property to multiply -5 by 1+2x.

$$-3x^{2}-7x+6=x\left(-x\right)+2x-5-10x$$

Combine 2x and -10x to get -8x.

$$-3x^{2}-7x+6=x\left(-x\right)-8x-5$$

Subtract x\left(-x\right) from both sides.

$$-3x^{2}-7x+6-x\left(-x\right)=-8x-5$$

Add 8x to both sides.

$$-3x^{2}-7x+6-x\left(-x\right)+8x=-5$$

Add 5 to both sides.

$$-3x^{2}-7x+6-x\left(-x\right)+8x+5=0$$

Multiply x and x to get x^{2}.

$$-3x^{2}-7x+6-x^{2}\left(-1\right)+8x+5=0$$

Multiply -1 and -1 to get 1.

$$-3x^{2}-7x+6+x^{2}+8x+5=0$$

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

$$-2x^{2}-7x+6+8x+5=0$$

Combine -7x and 8x to get x.

$$-2x^{2}+x+6+5=0$$

Add 6 and 5 to get 11.

$$-2x^{2}+x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\times 11}}{2\left(-2\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\times 11}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8\times 11}}{2\left(-2\right)}$$

Multiply 8 times 11.

$$x=\frac{-1±\sqrt{1+88}}{2\left(-2\right)}$$

Add 1 to 88.

$$x=\frac{-1±\sqrt{89}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1±\sqrt{89}}{-4}$$

Now solve the equation x=\frac{-1±\sqrt{89}}{-4} when ± is plus. Add -1 to \sqrt{89}.

$$x=\frac{\sqrt{89}-1}{-4}$$

Divide -1+\sqrt{89} by -4.

$$x=\frac{1-\sqrt{89}}{4}$$

Now solve the equation x=\frac{-1±\sqrt{89}}{-4} when ± is minus. Subtract \sqrt{89} from -1.

$$x=\frac{-\sqrt{89}-1}{-4}$$

Divide -1-\sqrt{89} by -4.

$$x=\frac{\sqrt{89}+1}{4}$$

The equation is now solved.

$$x=\frac{1-\sqrt{89}}{4} x=\frac{\sqrt{89}+1}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+3\right), the least common multiple of x+3,6-x-x^{2}.

$$-\left(x+3\right)\times 2x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

To find the opposite of x+3, find the opposite of each term.

$$\left(-x-3\right)\times 2x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply -x-3 by 2.

$$\left(-2x-6\right)x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply -2x-6 by x.

$$-2x^{2}-6x+\left(x-2\right)\left(x+3\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

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

$$-2x^{2}-6x+\left(x^{2}+x-6\right)\left(-1\right)=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply x^{2}+x-6 by -1.

$$-2x^{2}-6x-x^{2}-x+6=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

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

$$-3x^{2}-6x-x+6=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Combine -6x and -x to get -7x.

$$-3x^{2}-7x+6=\left(x-2\right)\left(-x\right)-5\left(1+2x\right)$$

Use the distributive property to multiply x-2 by -x.

$$-3x^{2}-7x+6=x\left(-x\right)-2\left(-x\right)-5\left(1+2x\right)$$

Multiply -2 and -1 to get 2.

$$-3x^{2}-7x+6=x\left(-x\right)+2x-5\left(1+2x\right)$$

Use the distributive property to multiply -5 by 1+2x.

$$-3x^{2}-7x+6=x\left(-x\right)+2x-5-10x$$

Combine 2x and -10x to get -8x.

$$-3x^{2}-7x+6=x\left(-x\right)-8x-5$$

Subtract x\left(-x\right) from both sides.

$$-3x^{2}-7x+6-x\left(-x\right)=-8x-5$$

Add 8x to both sides.

$$-3x^{2}-7x+6-x\left(-x\right)+8x=-5$$

Multiply x and x to get x^{2}.

$$-3x^{2}-7x+6-x^{2}\left(-1\right)+8x=-5$$

Multiply -1 and -1 to get 1.

$$-3x^{2}-7x+6+x^{2}+8x=-5$$

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

$$-2x^{2}-7x+6+8x=-5$$

Combine -7x and 8x to get x.

$$-2x^{2}+x+6=-5$$

Subtract 6 from both sides.

$$-2x^{2}+x=-5-6$$

Subtract 6 from -5 to get -11.

$$-2x^{2}+x=-11$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1}{-2}x=-\frac{11}{-2}$$

Divide 1 by -2.

$$x^{2}-\frac{1}{2}x=-\frac{11}{-2}$$

Divide -11 by -2.

$$x^{2}-\frac{1}{2}x=\frac{11}{2}$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{11}{2}+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{11}{2}+\frac{1}{16}$$

Add \frac{11}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{89}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{89}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{89}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{89}}{4} x-\frac{1}{4}=-\frac{\sqrt{89}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{89}+1}{4} x=\frac{1-\sqrt{89}}{4}$$