Solve 0.2=0.24+0.04x/x-0.01x

$$0.2 = \frac { 0.24 + 0.04 x } { x } - 0.01 x$$

Solve for x

$$x=2\sqrt{22}-8\approx 1.38083152$$

$$x=-2\sqrt{22}-8\approx -17.38083152$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$0.2x=0.24+0.04x-0.01xx$$

Multiply x and x to get x^{2}.

$$0.2x=0.24+0.04x-0.01x^{2}$$

Subtract 0.24 from both sides.

$$0.2x-0.24=0.04x-0.01x^{2}$$

Subtract 0.04x from both sides.

$$0.2x-0.24-0.04x=-0.01x^{2}$$

Combine 0.2x and -0.04x to get 0.16x.

$$0.16x-0.24=-0.01x^{2}$$

Add 0.01x^{2} to both sides.

$$0.16x-0.24+0.01x^{2}=0$$

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.

$$0.01x^{2}+0.16x-0.24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.01 for a, 0.16 for b, and -0.24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.16±\sqrt{0.16^{2}-4\times 0.01\left(-0.24\right)}}{2\times 0.01}$$

Square 0.16 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.16±\sqrt{0.0256-4\times 0.01\left(-0.24\right)}}{2\times 0.01}$$

Multiply -4 times 0.01.

$$x=\frac{-0.16±\sqrt{0.0256-0.04\left(-0.24\right)}}{2\times 0.01}$$

Multiply -0.04 times -0.24 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.16±\sqrt{\frac{16+6}{625}}}{2\times 0.01}$$

Add 0.0256 to 0.0096 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.16±\sqrt{0.0352}}{2\times 0.01}$$

Take the square root of 0.0352.

$$x=\frac{-0.16±\frac{\sqrt{22}}{25}}{2\times 0.01}$$

Multiply 2 times 0.01.

$$x=\frac{-0.16±\frac{\sqrt{22}}{25}}{0.02}$$

Now solve the equation x=\frac{-0.16±\frac{\sqrt{22}}{25}}{0.02} when ± is plus. Add -0.16 to \frac{\sqrt{22}}{25}.

$$x=\frac{\sqrt{22}-4}{0.02\times 25}$$

Divide \frac{-4+\sqrt{22}}{25} by 0.02 by multiplying \frac{-4+\sqrt{22}}{25} by the reciprocal of 0.02.

$$x=2\sqrt{22}-8$$

Now solve the equation x=\frac{-0.16±\frac{\sqrt{22}}{25}}{0.02} when ± is minus. Subtract \frac{\sqrt{22}}{25} from -0.16.

$$x=\frac{-\sqrt{22}-4}{0.02\times 25}$$

Divide \frac{-4-\sqrt{22}}{25} by 0.02 by multiplying \frac{-4-\sqrt{22}}{25} by the reciprocal of 0.02.

$$x=-2\sqrt{22}-8$$

The equation is now solved.

$$x=2\sqrt{22}-8 x=-2\sqrt{22}-8$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$0.2x=0.24+0.04x-0.01xx$$

Multiply x and x to get x^{2}.

$$0.2x=0.24+0.04x-0.01x^{2}$$

Subtract 0.04x from both sides.

$$0.2x-0.04x=0.24-0.01x^{2}$$

Combine 0.2x and -0.04x to get 0.16x.

$$0.16x=0.24-0.01x^{2}$$

Add 0.01x^{2} to both sides.

$$0.16x+0.01x^{2}=0.24$$

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.

$$0.01x^{2}+0.16x=0.24$$

Multiply both sides by 100.

$$\frac{0.01x^{2}+0.16x}{0.01}=\frac{0.24}{0.01}$$

Dividing by 0.01 undoes the multiplication by 0.01.

$$x^{2}+\frac{0.16}{0.01}x=\frac{0.24}{0.01}$$

Divide 0.16 by 0.01 by multiplying 0.16 by the reciprocal of 0.01.

$$x^{2}+16x=\frac{0.24}{0.01}$$

Divide 0.24 by 0.01 by multiplying 0.24 by the reciprocal of 0.01.

$$x^{2}+16x=24$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=24+8^{2}$$

Square 8.

$$x^{2}+16x+64=24+64$$

Add 24 to 64.

$$x^{2}+16x+64=88$$

Factor x^{2}+16x+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}.

$$\left(x+8\right)^{2}=88$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{88}$$

Simplify.

$$x+8=2\sqrt{22} x+8=-2\sqrt{22}$$

Subtract 8 from both sides of the equation.

$$x=2\sqrt{22}-8 x=-2\sqrt{22}-8$$