Solve 21/x+1=16/x-2-6/x

$$\frac { 21 } { x + 1 } = \frac { 16 } { x - 2 } - \frac { 6 } { x }$$

Solve for x

$$x=-\frac{2}{11}\approx -0.181818182$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right)\left(x+1\right), the least common multiple of x+1,x-2,x.

$$x\left(x-2\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$\left(x^{2}-2x\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Use the distributive property to multiply x^{2}-2x by 21.

$$21x^{2}-42x=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Use the distributive property to multiply x by x+1.

$$21x^{2}-42x=\left(x^{2}+x\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Use the distributive property to multiply x^{2}+x by 16.

$$21x^{2}-42x=16x^{2}+16x-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$21x^{2}-42x=16x^{2}+16x-\left(x^{2}-x-2\right)\times 6$$

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

$$21x^{2}-42x=16x^{2}+16x-\left(6x^{2}-6x-12\right)$$

To find the opposite of 6x^{2}-6x-12, find the opposite of each term.

$$21x^{2}-42x=16x^{2}+16x-6x^{2}+6x+12$$

Combine 16x^{2} and -6x^{2} to get 10x^{2}.

$$21x^{2}-42x=10x^{2}+16x+6x+12$$

Combine 16x and 6x to get 22x.

$$21x^{2}-42x=10x^{2}+22x+12$$

Subtract 10x^{2} from both sides.

$$21x^{2}-42x-10x^{2}=22x+12$$

Combine 21x^{2} and -10x^{2} to get 11x^{2}.

$$11x^{2}-42x=22x+12$$

Subtract 22x from both sides.

$$11x^{2}-42x-22x=12$$

Combine -42x and -22x to get -64x.

$$11x^{2}-64x=12$$

Subtract 12 from both sides.

$$11x^{2}-64x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, -64 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 11\left(-12\right)}}{2\times 11}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 11\left(-12\right)}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-64\right)±\sqrt{4096-44\left(-12\right)}}{2\times 11}$$

Multiply -44 times -12.

$$x=\frac{-\left(-64\right)±\sqrt{4096+528}}{2\times 11}$$

Add 4096 to 528.

$$x=\frac{-\left(-64\right)±\sqrt{4624}}{2\times 11}$$

Take the square root of 4624.

$$x=\frac{-\left(-64\right)±68}{2\times 11}$$

The opposite of -64 is 64.

$$x=\frac{64±68}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{64±68}{22}$$

Now solve the equation x=\frac{64±68}{22} when ± is plus. Add 64 to 68.

$$x=\frac{132}{22}$$

Divide 132 by 22.

$$x=6$$

Now solve the equation x=\frac{64±68}{22} when ± is minus. Subtract 68 from 64.

$$x=-\frac{4}{22}$$

Reduce the fraction \frac{-4}{22} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=6 x=-\frac{2}{11}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right)\left(x+1\right), the least common multiple of x+1,x-2,x.

$$x\left(x-2\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$\left(x^{2}-2x\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Use the distributive property to multiply x^{2}-2x by 21.

$$21x^{2}-42x=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Use the distributive property to multiply x by x+1.

$$21x^{2}-42x=\left(x^{2}+x\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Use the distributive property to multiply x^{2}+x by 16.

$$21x^{2}-42x=16x^{2}+16x-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$21x^{2}-42x=16x^{2}+16x-\left(x^{2}-x-2\right)\times 6$$

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

$$21x^{2}-42x=16x^{2}+16x-\left(6x^{2}-6x-12\right)$$

To find the opposite of 6x^{2}-6x-12, find the opposite of each term.

$$21x^{2}-42x=16x^{2}+16x-6x^{2}+6x+12$$

Combine 16x^{2} and -6x^{2} to get 10x^{2}.

$$21x^{2}-42x=10x^{2}+16x+6x+12$$

Combine 16x and 6x to get 22x.

$$21x^{2}-42x=10x^{2}+22x+12$$

Subtract 10x^{2} from both sides.

$$21x^{2}-42x-10x^{2}=22x+12$$

Combine 21x^{2} and -10x^{2} to get 11x^{2}.

$$11x^{2}-42x=22x+12$$

Subtract 22x from both sides.

$$11x^{2}-42x-22x=12$$

Combine -42x and -22x to get -64x.

$$11x^{2}-64x=12$$

Divide both sides by 11.

$$\frac{11x^{2}-64x}{11}=\frac{12}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}-\frac{64}{11}x=\frac{12}{11}$$

Divide -\frac{64}{11}, the coefficient of the x term, by 2 to get -\frac{32}{11}. Then add the square of -\frac{32}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{64}{11}x+\left(-\frac{32}{11}\right)^{2}=\frac{12}{11}+\left(-\frac{32}{11}\right)^{2}$$

Square -\frac{32}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{64}{11}x+\frac{1024}{121}=\frac{12}{11}+\frac{1024}{121}$$

Add \frac{12}{11} to \frac{1024}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{64}{11}x+\frac{1024}{121}=\frac{1156}{121}$$

Factor x^{2}-\frac{64}{11}x+\frac{1024}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32}{11}\right)^{2}=\frac{1156}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{32}{11}\right)^{2}}=\sqrt{\frac{1156}{121}}$$

Simplify.

$$x-\frac{32}{11}=\frac{34}{11} x-\frac{32}{11}=-\frac{34}{11}$$

Add \frac{32}{11} to both sides of the equation.

$$x=6 x=-\frac{2}{11}$$