Solve left(2-2xright)^2/(2+x)=0.394

$$\frac{ { \left(2-2x \right) }^{ 2 } }{ (2+x) } =0.394$$

Solve for x

$$x = \frac{\sqrt{4766809} + 4197}{4000} \approx 1.595075579$$

$$x=\frac{4197-\sqrt{4766809}}{4000}\approx 0.503424421$$

Steps Using the Quadratic Formula

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

$$\left(2-2x\right)^{2}=0.394\left(x+2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-2x\right)^{2}.

$$4-8x+4x^{2}=0.394\left(x+2\right)$$

Use the distributive property to multiply 0.394 by x+2.

$$4-8x+4x^{2}=0.394x+0.788$$

Subtract 0.394x from both sides.

$$4-8x+4x^{2}-0.394x=0.788$$

Combine -8x and -0.394x to get -8.394x.

$$4-8.394x+4x^{2}=0.788$$

Subtract 0.788 from both sides.

$$4-8.394x+4x^{2}-0.788=0$$

Subtract 0.788 from 4 to get 3.212.

$$3.212-8.394x+4x^{2}=0$$

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.

$$4x^{2}-8.394x+3.212=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -8.394 for b, and 3.212 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8.394\right)±\sqrt{\left(-8.394\right)^{2}-4\times 4\times 3.212}}{2\times 4}$$

Square -8.394 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-8.394\right)±\sqrt{70.459236-4\times 4\times 3.212}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-8.394\right)±\sqrt{70.459236-16\times 3.212}}{2\times 4}$$

Multiply -16 times 3.212.

$$x=\frac{-\left(-8.394\right)±\sqrt{70.459236-51.392}}{2\times 4}$$

Add 70.459236 to -51.392 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-8.394\right)±\sqrt{19.067236}}{2\times 4}$$

Take the square root of 19.067236.

$$x=\frac{-\left(-8.394\right)±\frac{\sqrt{4766809}}{500}}{2\times 4}$$

The opposite of -8.394 is 8.394.

$$x=\frac{8.394±\frac{\sqrt{4766809}}{500}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{8.394±\frac{\sqrt{4766809}}{500}}{8}$$

Now solve the equation x=\frac{8.394±\frac{\sqrt{4766809}}{500}}{8} when ± is plus. Add 8.394 to \frac{\sqrt{4766809}}{500}.

$$x=\frac{\sqrt{4766809}+4197}{8\times 500}$$

Divide \frac{4197+\sqrt{4766809}}{500} by 8.

$$x=\frac{\sqrt{4766809}+4197}{4000}$$

Now solve the equation x=\frac{8.394±\frac{\sqrt{4766809}}{500}}{8} when ± is minus. Subtract \frac{\sqrt{4766809}}{500} from 8.394.

$$x=\frac{4197-\sqrt{4766809}}{8\times 500}$$

Divide \frac{4197-\sqrt{4766809}}{500} by 8.

$$x=\frac{4197-\sqrt{4766809}}{4000}$$

The equation is now solved.

$$x=\frac{\sqrt{4766809}+4197}{4000} x=\frac{4197-\sqrt{4766809}}{4000}$$

Steps for Completing the Square

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

$$\left(2-2x\right)^{2}=0.394\left(x+2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-2x\right)^{2}.

$$4-8x+4x^{2}=0.394\left(x+2\right)$$

Use the distributive property to multiply 0.394 by x+2.

$$4-8x+4x^{2}=0.394x+0.788$$

Subtract 0.394x from both sides.

$$4-8x+4x^{2}-0.394x=0.788$$

Combine -8x and -0.394x to get -8.394x.

$$4-8.394x+4x^{2}=0.788$$

Subtract 4 from both sides.

$$-8.394x+4x^{2}=0.788-4$$

Subtract 4 from 0.788 to get -3.212.

$$-8.394x+4x^{2}=-3.212$$

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.

$$4x^{2}-8.394x=-3.212$$

Divide both sides by 4.

$$\frac{4x^{2}-8.394x}{4}=-\frac{3.212}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{8.394}{4}\right)x=-\frac{3.212}{4}$$

Divide -8.394 by 4.

$$x^{2}-2.0985x=-\frac{3.212}{4}$$

Divide -3.212 by 4.

$$x^{2}-2.0985x=-0.803$$

Divide -2.0985, the coefficient of the x term, by 2 to get -1.04925. Then add the square of -1.04925 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2.0985x+\left(-1.04925\right)^{2}=-0.803+\left(-1.04925\right)^{2}$$

Square -1.04925 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-2.0985x+1.1009255625=-0.803+1.1009255625$$

Add -0.803 to 1.1009255625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-2.0985x+1.1009255625=0.2979255625$$

Factor x^{2}-2.0985x+1.1009255625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1.04925\right)^{2}=0.2979255625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1.04925\right)^{2}}=\sqrt{0.2979255625}$$

Simplify.

$$x-1.04925=\frac{\sqrt{4766809}}{4000} x-1.04925=-\frac{\sqrt{4766809}}{4000}$$

Add 1.04925 to both sides of the equation.

$$x=\frac{\sqrt{4766809}+4197}{4000} x=\frac{4197-\sqrt{4766809}}{4000}$$