Solve 4/x+4/12-2x=1

$$\frac { 4 } { x } + \frac { 4 } { 12 - 2 x } = 1$$

Solve for x (complex solution)

$$x=4+2\sqrt{2}i\approx 4+2.828427125i$$

$$x=-2\sqrt{2}i+4\approx 4-2.828427125i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-6\right), the least common multiple of x,12-2x.

$$\left(2x-12\right)\times 4-x\times 4=2x\left(x-6\right)$$

Use the distributive property to multiply 2x-12 by 4.

$$8x-48-x\times 4=2x\left(x-6\right)$$

Multiply -1 and 4 to get -4.

$$8x-48-4x=2x\left(x-6\right)$$

Combine 8x and -4x to get 4x.

$$4x-48=2x\left(x-6\right)$$

Use the distributive property to multiply 2x by x-6.

$$4x-48=2x^{2}-12x$$

Subtract 2x^{2} from both sides.

$$4x-48-2x^{2}=-12x$$

Add 12x to both sides.

$$4x-48-2x^{2}+12x=0$$

Combine 4x and 12x to get 16x.

$$16x-48-2x^{2}=0$$

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.

$$-2x^{2}+16x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 16 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-2\right)\left(-48\right)}}{2\left(-2\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-2\right)\left(-48\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-16±\sqrt{256+8\left(-48\right)}}{2\left(-2\right)}$$

Multiply 8 times -48.

$$x=\frac{-16±\sqrt{256-384}}{2\left(-2\right)}$$

Add 256 to -384.

$$x=\frac{-16±\sqrt{-128}}{2\left(-2\right)}$$

Take the square root of -128.

$$x=\frac{-16±8\sqrt{2}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-16±8\sqrt{2}i}{-4}$$

Now solve the equation x=\frac{-16±8\sqrt{2}i}{-4} when ± is plus. Add -16 to 8i\sqrt{2}.

$$x=\frac{-16+2^{\frac{7}{2}}i}{-4}$$

Divide -16+i\times 2^{\frac{7}{2}} by -4.

$$x=-2\sqrt{2}i+4$$

Now solve the equation x=\frac{-16±8\sqrt{2}i}{-4} when ± is minus. Subtract 8i\sqrt{2} from -16.

$$x=\frac{-2^{\frac{7}{2}}i-16}{-4}$$

Divide -16-i\times 2^{\frac{7}{2}} by -4.

$$x=4+2\sqrt{2}i$$

The equation is now solved.

$$x=-2\sqrt{2}i+4 x=4+2\sqrt{2}i$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-6\right), the least common multiple of x,12-2x.

$$\left(2x-12\right)\times 4-x\times 4=2x\left(x-6\right)$$

Use the distributive property to multiply 2x-12 by 4.

$$8x-48-x\times 4=2x\left(x-6\right)$$

Multiply -1 and 4 to get -4.

$$8x-48-4x=2x\left(x-6\right)$$

Combine 8x and -4x to get 4x.

$$4x-48=2x\left(x-6\right)$$

Use the distributive property to multiply 2x by x-6.

$$4x-48=2x^{2}-12x$$

Subtract 2x^{2} from both sides.

$$4x-48-2x^{2}=-12x$$

Add 12x to both sides.

$$4x-48-2x^{2}+12x=0$$

Combine 4x and 12x to get 16x.

$$16x-48-2x^{2}=0$$

Add 48 to both sides. Anything plus zero gives itself.

$$16x-2x^{2}=48$$

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.

$$-2x^{2}+16x=48$$

Divide both sides by -2.

$$\frac{-2x^{2}+16x}{-2}=\frac{48}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{16}{-2}x=\frac{48}{-2}$$

Divide 16 by -2.

$$x^{2}-8x=\frac{48}{-2}$$

Divide 48 by -2.

$$x^{2}-8x=-24$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-24+\left(-4\right)^{2}$$

Square -4.

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

Add -24 to 16.

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

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-4=2\sqrt{2}i x-4=-2\sqrt{2}i$$

Add 4 to both sides of the equation.

$$x=4+2\sqrt{2}i x=-2\sqrt{2}i+4$$