Solve -4/2x+0.5x/2=4x-1/4x

$$- \frac { 4 } { 2 x } + \frac { 0.5 x } { 2 } = \frac { 4 x - 1 } { 4 x }$$

Solve for x

$$x=\sqrt{11}+2\approx 5.31662479$$

$$x=2-\sqrt{11}\approx -1.31662479$$

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 4x, the least common multiple of 2x,2,4x.

$$-2\times 4+2x\times 0.5x=4x-1$$

Multiply -2 and 4 to get -8.

$$-8+2x\times 0.5x=4x-1$$

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

$$-8+2x^{2}\times 0.5=4x-1$$

Multiply 2 and 0.5 to get 1.

$$-8+x^{2}=4x-1$$

Subtract 4x from both sides.

$$-8+x^{2}-4x=-1$$

Add 1 to both sides.

$$-8+x^{2}-4x+1=0$$

Add -8 and 1 to get -7.

$$-7+x^{2}-4x=0$$

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.

$$x^{2}-4x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-7\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$x=\frac{-\left(-4\right)±\sqrt{16+28}}{2}$$

Add 16 to 28.

$$x=\frac{-\left(-4\right)±\sqrt{44}}{2}$$

Take the square root of 44.

$$x=\frac{-\left(-4\right)±2\sqrt{11}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{11}}{2}$$

Now solve the equation x=\frac{4±2\sqrt{11}}{2} when ± is plus. Add 4 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}+4}{2}$$

Divide 4+2\sqrt{11} by 2.

$$x=\sqrt{11}+2$$

Now solve the equation x=\frac{4±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from 4.

$$x=\frac{4-2\sqrt{11}}{2}$$

Divide 4-2\sqrt{11} by 2.

$$x=2-\sqrt{11}$$

The equation is now solved.

$$x=\sqrt{11}+2 x=2-\sqrt{11}$$

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 4x, the least common multiple of 2x,2,4x.

$$-2\times 4+2x\times 0.5x=4x-1$$

Multiply -2 and 4 to get -8.

$$-8+2x\times 0.5x=4x-1$$

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

$$-8+2x^{2}\times 0.5=4x-1$$

Multiply 2 and 0.5 to get 1.

$$-8+x^{2}=4x-1$$

Subtract 4x from both sides.

$$-8+x^{2}-4x=-1$$

Add 8 to both sides.

$$x^{2}-4x=-1+8$$

Add -1 and 8 to get 7.

$$x^{2}-4x=7$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=7+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=7+4$$

Add 7 to 4.

$$x^{2}-4x+4=11$$

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}.

$$\left(x-2\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{11}$$

Simplify.

$$x-2=\sqrt{11} x-2=-\sqrt{11}$$

Add 2 to both sides of the equation.

$$x=\sqrt{11}+2 x=2-\sqrt{11}$$