Solve 1div(4+x)=0.2x

$$1 \div (4+x)=0.2x$$

Solve for x

$$x=-5$$

$$x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$1=0.2x\left(x+4\right)$$

Use the distributive property to multiply 0.2x by x+4.

$$1=0.2x^{2}+0.8x$$

Swap sides so that all variable terms are on the left hand side.

$$0.2x^{2}+0.8x=1$$

Subtract 1 from both sides.

$$0.2x^{2}+0.8x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.2 for a, 0.8 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.8±\sqrt{0.8^{2}-4\times 0.2\left(-1\right)}}{2\times 0.2}$$

Square 0.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.8±\sqrt{0.64-4\times 0.2\left(-1\right)}}{2\times 0.2}$$

Multiply -4 times 0.2.

$$x=\frac{-0.8±\sqrt{0.64-0.8\left(-1\right)}}{2\times 0.2}$$

Multiply -0.8 times -1.

$$x=\frac{-0.8±\sqrt{0.64+0.8}}{2\times 0.2}$$

Add 0.64 to 0.8 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.8±\sqrt{1.44}}{2\times 0.2}$$

Take the square root of 1.44.

$$x=\frac{-0.8±\frac{6}{5}}{2\times 0.2}$$

Multiply 2 times 0.2.

$$x=\frac{-0.8±\frac{6}{5}}{0.4}$$

Now solve the equation x=\frac{-0.8±\frac{6}{5}}{0.4} when ± is plus. Add -0.8 to \frac{6}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{2}{5}}{0.4}$$

Divide \frac{2}{5} by 0.4 by multiplying \frac{2}{5} by the reciprocal of 0.4.

$$x=1$$

Now solve the equation x=\frac{-0.8±\frac{6}{5}}{0.4} when ± is minus. Subtract \frac{6}{5} from -0.8 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -2 by 0.4 by multiplying -2 by the reciprocal of 0.4.

$$x=-5$$

The equation is now solved.

$$x=1 x=-5$$

Steps for Completing the Square

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$1=0.2x\left(x+4\right)$$

Use the distributive property to multiply 0.2x by x+4.

$$1=0.2x^{2}+0.8x$$

Swap sides so that all variable terms are on the left hand side.

$$0.2x^{2}+0.8x=1$$

Multiply both sides by 5.

$$\frac{0.2x^{2}+0.8x}{0.2}=\frac{1}{0.2}$$

Dividing by 0.2 undoes the multiplication by 0.2.

$$x^{2}+\frac{0.8}{0.2}x=\frac{1}{0.2}$$

Divide 0.8 by 0.2 by multiplying 0.8 by the reciprocal of 0.2.

$$x^{2}+4x=\frac{1}{0.2}$$

Divide 1 by 0.2 by multiplying 1 by the reciprocal of 0.2.

$$x^{2}+4x=5$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=5+2^{2}$$

Square 2.

$$x^{2}+4x+4=5+4$$

Add 5 to 4.

$$x^{2}+4x+4=9$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+2=3 x+2=-3$$

Subtract 2 from both sides of the equation.

$$x=1 x=-5$$