Solve x-2/x+2/2/frac{1-2x{0.4}}=0.4

$$\frac { \frac { \frac { x - 2 } { x + 2 } } { 2 } } { \frac { 1 - 2 x } { 0.4 } } = 0.4$$

Solve for x

$$x=\frac{\sqrt{145}-7}{8}\approx 0.630199322$$

$$x=\frac{-\sqrt{145}-7}{8}\approx -2.380199322$$

Steps Using the Quadratic Formula

Divide \frac{\frac{x-2}{x+2}}{2} by \frac{1-2x}{0.4} by multiplying \frac{\frac{x-2}{x+2}}{2} by the reciprocal of \frac{1-2x}{0.4}.

$$\frac{\frac{x-2}{x+2}\times 0.4}{2\left(1-2x\right)}=0.4$$

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

$$\frac{\frac{x-2}{x+2}\times 0.4}{2-4x}=0.4$$

Subtract 0.4 from both sides.

$$\frac{\frac{x-2}{x+2}\times 0.4}{2-4x}-0.4=0$$

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

$$\frac{x-2}{x+2}\times 0.4+2\left(-2x+1\right)\left(-0.4\right)=0$$

Reorder the terms.

$$-0.4\times 2\left(-2x+1\right)+0.4\times \frac{x-2}{x+2}=0$$

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$-0.4\times 2\left(-2x+1\right)\left(x+2\right)+0.4\left(x-2\right)=0$$

Multiply -0.4 and 2 to get -0.8.

$$-0.8\left(-2x+1\right)\left(x+2\right)+0.4\left(x-2\right)=0$$

Use the distributive property to multiply -0.8 by -2x+1.

$$\left(1.6x-0.8\right)\left(x+2\right)+0.4\left(x-2\right)=0$$

Use the distributive property to multiply 1.6x-0.8 by x+2 and combine like terms.

$$1.6x^{2}+2.4x-1.6+0.4\left(x-2\right)=0$$

Use the distributive property to multiply 0.4 by x-2.

$$1.6x^{2}+2.4x-1.6+0.4x-0.8=0$$

Combine 2.4x and 0.4x to get 2.8x.

$$1.6x^{2}+2.8x-1.6-0.8=0$$

Subtract 0.8 from -1.6 to get -2.4.

$$1.6x^{2}+2.8x-2.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.6 for a, 2.8 for b, and -2.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2.8±\sqrt{2.8^{2}-4\times 1.6\left(-2.4\right)}}{2\times 1.6}$$

Square 2.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-2.8±\sqrt{7.84-4\times 1.6\left(-2.4\right)}}{2\times 1.6}$$

Multiply -4 times 1.6.

$$x=\frac{-2.8±\sqrt{7.84-6.4\left(-2.4\right)}}{2\times 1.6}$$

Multiply -6.4 times -2.4 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-2.8±\sqrt{\frac{196+384}{25}}}{2\times 1.6}$$

Add 7.84 to 15.36 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-2.8±\sqrt{23.2}}{2\times 1.6}$$

Take the square root of 23.2.

$$x=\frac{-2.8±\frac{2\sqrt{145}}{5}}{2\times 1.6}$$

Multiply 2 times 1.6.

$$x=\frac{-2.8±\frac{2\sqrt{145}}{5}}{3.2}$$

Now solve the equation x=\frac{-2.8±\frac{2\sqrt{145}}{5}}{3.2} when ± is plus. Add -2.8 to \frac{2\sqrt{145}}{5}.

$$x=\frac{2\sqrt{145}-14}{3.2\times 5}$$

Divide \frac{-14+2\sqrt{145}}{5} by 3.2 by multiplying \frac{-14+2\sqrt{145}}{5} by the reciprocal of 3.2.

$$x=\frac{\sqrt{145}-7}{8}$$

Now solve the equation x=\frac{-2.8±\frac{2\sqrt{145}}{5}}{3.2} when ± is minus. Subtract \frac{2\sqrt{145}}{5} from -2.8.

$$x=\frac{-2\sqrt{145}-14}{3.2\times 5}$$

Divide \frac{-14-2\sqrt{145}}{5} by 3.2 by multiplying \frac{-14-2\sqrt{145}}{5} by the reciprocal of 3.2.

$$x=\frac{-\sqrt{145}-7}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{145}-7}{8} x=\frac{-\sqrt{145}-7}{8}$$

Steps for Completing the Square

Divide \frac{\frac{x-2}{x+2}}{2} by \frac{1-2x}{0.4} by multiplying \frac{\frac{x-2}{x+2}}{2} by the reciprocal of \frac{1-2x}{0.4}.

$$\frac{\frac{x-2}{x+2}\times 0.4}{2\left(1-2x\right)}=0.4$$

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

$$\frac{\frac{x-2}{x+2}\times 0.4}{2-4x}=0.4$$

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

$$\frac{x-2}{x+2}\times 0.4=0.8\left(-2x+1\right)$$

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$\left(x-2\right)\times 0.4=0.8\left(-2x+1\right)\left(x+2\right)$$

Use the distributive property to multiply x-2 by 0.4.

$$0.4x-0.8=0.8\left(-2x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 0.8 by -2x+1.

$$0.4x-0.8=\left(-1.6x+0.8\right)\left(x+2\right)$$

Use the distributive property to multiply -1.6x+0.8 by x+2 and combine like terms.

$$0.4x-0.8=-1.6x^{2}-2.4x+1.6$$

Add 1.6x^{2} to both sides.

$$0.4x-0.8+1.6x^{2}=-2.4x+1.6$$

Add 2.4x to both sides.

$$0.4x-0.8+1.6x^{2}+2.4x=1.6$$

Combine 0.4x and 2.4x to get 2.8x.

$$2.8x-0.8+1.6x^{2}=1.6$$

Add 0.8 to both sides.

$$2.8x+1.6x^{2}=1.6+0.8$$

Add 1.6 and 0.8 to get 2.4.

$$2.8x+1.6x^{2}=2.4$$

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.

$$1.6x^{2}+2.8x=2.4$$

Divide both sides of the equation by 1.6, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.6x^{2}+2.8x}{1.6}=\frac{2.4}{1.6}$$

Dividing by 1.6 undoes the multiplication by 1.6.

$$x^{2}+\frac{2.8}{1.6}x=\frac{2.4}{1.6}$$

Divide 2.8 by 1.6 by multiplying 2.8 by the reciprocal of 1.6.

$$x^{2}+1.75x=\frac{2.4}{1.6}$$

Divide 2.4 by 1.6 by multiplying 2.4 by the reciprocal of 1.6.

$$x^{2}+1.75x=1.5$$

Divide 1.75, the coefficient of the x term, by 2 to get 0.875. Then add the square of 0.875 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1.75x+0.875^{2}=1.5+0.875^{2}$$

Square 0.875 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+1.75x+0.765625=1.5+0.765625$$

Add 1.5 to 0.765625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+1.75x+0.765625=2.265625$$

Factor x^{2}+1.75x+0.765625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.875\right)^{2}=2.265625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.875\right)^{2}}=\sqrt{2.265625}$$

Simplify.

$$x+0.875=\frac{\sqrt{145}}{8} x+0.875=-\frac{\sqrt{145}}{8}$$

Subtract 0.875 from both sides of the equation.

$$x=\frac{\sqrt{145}-7}{8} x=\frac{-\sqrt{145}-7}{8}$$