Solve x(x+1)/(2-x)(2+x)=4

$$\frac { x ( x + 1 ) } { ( 2 - x ) ( 2 + x ) } = 4$$

Solve for x

$$x = \frac{\sqrt{321} - 1}{10} \approx 1.691647287$$

$$x=\frac{-\sqrt{321}-1}{10}\approx -1.891647287$$

Steps Using the Quadratic Formula

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

$$x\left(x+1\right)=4\left(x-2\right)\left(-x-2\right)$$

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

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

Use the distributive property to multiply 4 by x-2.

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

Use the distributive property to multiply 4x-8 by -x-2 and combine like terms.

$$x^{2}+x=-4x^{2}+16$$

Add 4x^{2} to both sides.

$$x^{2}+x+4x^{2}=16$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}+x=16$$

Subtract 16 from both sides.

$$5x^{2}+x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 1 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 5\left(-16\right)}}{2\times 5}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 5\left(-16\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-1±\sqrt{1-20\left(-16\right)}}{2\times 5}$$

Multiply -20 times -16.

$$x=\frac{-1±\sqrt{1+320}}{2\times 5}$$

Add 1 to 320.

$$x=\frac{-1±\sqrt{321}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-1±\sqrt{321}}{10}$$

Now solve the equation x=\frac{-1±\sqrt{321}}{10} when ± is plus. Add -1 to \sqrt{321}.

$$x=\frac{\sqrt{321}-1}{10}$$

Now solve the equation x=\frac{-1±\sqrt{321}}{10} when ± is minus. Subtract \sqrt{321} from -1.

$$x=\frac{-\sqrt{321}-1}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{321}-1}{10} x=\frac{-\sqrt{321}-1}{10}$$

Steps for Completing the Square

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

$$x\left(x+1\right)=4\left(x-2\right)\left(-x-2\right)$$

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

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

Use the distributive property to multiply 4 by x-2.

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

Use the distributive property to multiply 4x-8 by -x-2 and combine like terms.

$$x^{2}+x=-4x^{2}+16$$

Add 4x^{2} to both sides.

$$x^{2}+x+4x^{2}=16$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}+x=16$$

Divide both sides by 5.

$$\frac{5x^{2}+x}{5}=\frac{16}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{1}{5}x=\frac{16}{5}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\frac{16}{5}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{16}{5}+\frac{1}{100}$$

Add \frac{16}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{321}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{321}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{321}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{\sqrt{321}}{10} x+\frac{1}{10}=-\frac{\sqrt{321}}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=\frac{\sqrt{321}-1}{10} x=\frac{-\sqrt{321}-1}{10}$$