Solve 2x-1/2x+1-2x+1/2x-1=-22/3

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

Solve for x

$$x=1$$

$$x=-\frac{1}{4}=-0.25$$

Steps Using the Quadratic Formula

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

$$\left(6x-3\right)\left(2x-1\right)-\left(6x+3\right)\left(2x+1\right)=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Use the distributive property to multiply 6x-3 by 2x-1 and combine like terms.

$$12x^{2}-12x+3-\left(6x+3\right)\left(2x+1\right)=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Use the distributive property to multiply 6x+3 by 2x+1 and combine like terms.

$$12x^{2}-12x+3-\left(12x^{2}+12x+3\right)=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

To find the opposite of 12x^{2}+12x+3, find the opposite of each term.

$$12x^{2}-12x+3-12x^{2}-12x-3=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Combine 12x^{2} and -12x^{2} to get 0.

$$-12x+3-12x-3=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Combine -12x and -12x to get -24x.

$$-24x+3-3=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Subtract 3 from 3 to get 0.

$$-24x=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Multiply 2 and 3 to get 6.

$$-24x=-\left(4x^{2}-1\right)\left(6+2\right)$$

Add 6 and 2 to get 8.

$$-24x=-\left(4x^{2}-1\right)\times 8$$

Use the distributive property to multiply 4x^{2}-1 by 8.

$$-24x=-\left(32x^{2}-8\right)$$

To find the opposite of 32x^{2}-8, find the opposite of each term.

$$-24x=-32x^{2}+8$$

Add 32x^{2} to both sides.

$$-24x+32x^{2}=8$$

Subtract 8 from both sides.

$$-24x+32x^{2}-8=0$$

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.

$$32x^{2}-24x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 32 for a, -24 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 32\left(-8\right)}}{2\times 32}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 32\left(-8\right)}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-24\right)±\sqrt{576-128\left(-8\right)}}{2\times 32}$$

Multiply -128 times -8.

$$x=\frac{-\left(-24\right)±\sqrt{576+1024}}{2\times 32}$$

Add 576 to 1024.

$$x=\frac{-\left(-24\right)±\sqrt{1600}}{2\times 32}$$

Take the square root of 1600.

$$x=\frac{-\left(-24\right)±40}{2\times 32}$$

The opposite of -24 is 24.

$$x=\frac{24±40}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{24±40}{64}$$

Now solve the equation x=\frac{24±40}{64} when ± is plus. Add 24 to 40.

$$x=\frac{64}{64}$$

Divide 64 by 64.

$$x=1$$

Now solve the equation x=\frac{24±40}{64} when ± is minus. Subtract 40 from 24.

$$x=-\frac{16}{64}$$

Reduce the fraction \frac{-16}{64} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$\left(6x-3\right)\left(2x-1\right)-\left(6x+3\right)\left(2x+1\right)=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Use the distributive property to multiply 6x-3 by 2x-1 and combine like terms.

$$12x^{2}-12x+3-\left(6x+3\right)\left(2x+1\right)=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Use the distributive property to multiply 6x+3 by 2x+1 and combine like terms.

$$12x^{2}-12x+3-\left(12x^{2}+12x+3\right)=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

To find the opposite of 12x^{2}+12x+3, find the opposite of each term.

$$12x^{2}-12x+3-12x^{2}-12x-3=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Combine 12x^{2} and -12x^{2} to get 0.

$$-12x+3-12x-3=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Combine -12x and -12x to get -24x.

$$-24x+3-3=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Subtract 3 from 3 to get 0.

$$-24x=-\left(4x^{2}-1\right)\left(2\times 3+2\right)$$

Multiply 2 and 3 to get 6.

$$-24x=-\left(4x^{2}-1\right)\left(6+2\right)$$

Add 6 and 2 to get 8.

$$-24x=-\left(4x^{2}-1\right)\times 8$$

Use the distributive property to multiply 4x^{2}-1 by 8.

$$-24x=-\left(32x^{2}-8\right)$$

To find the opposite of 32x^{2}-8, find the opposite of each term.

$$-24x=-32x^{2}+8$$

Add 32x^{2} to both sides.

$$-24x+32x^{2}=8$$

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.

$$32x^{2}-24x=8$$

Divide both sides by 32.

$$\frac{32x^{2}-24x}{32}=\frac{8}{32}$$

Dividing by 32 undoes the multiplication by 32.

$$x^{2}+\left(-\frac{24}{32}\right)x=\frac{8}{32}$$

Reduce the fraction \frac{-24}{32} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{3}{4}x=\frac{8}{32}$$

Reduce the fraction \frac{8}{32} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{3}{4}x=\frac{1}{4}$$

Divide -\frac{3}{4}, the coefficient of the x term, by 2 to get -\frac{3}{8}. Then add the square of -\frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=\frac{1}{4}+\left(-\frac{3}{8}\right)^{2}$$

Square -\frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{1}{4}+\frac{9}{64}$$

Add \frac{1}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{25}{64}$$

Factor x^{2}-\frac{3}{4}x+\frac{9}{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}.

$$\left(x-\frac{3}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x-\frac{3}{8}=\frac{5}{8} x-\frac{3}{8}=-\frac{5}{8}$$

Add \frac{3}{8} to both sides of the equation.

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