Solve 6/x^2-12/x=3

$$\frac { 6 } { x ^ { 2 } } - \frac { 12 } { x } = 3$$

Solve for x (complex solution)

$$x=\sqrt{6}-2\approx 0.449489743$$

$$x=-\left(\sqrt{6}+2\right)\approx -4.449489743$$

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^{2}, the least common multiple of x^{2},x.

$$6-x\times 12=3x^{2}$$

Subtract 3x^{2} from both sides.

$$6-x\times 12-3x^{2}=0$$

Multiply -1 and 12 to get -12.

$$6-12x-3x^{2}=0$$

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.

$$-3x^{2}-12x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -12 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-12\right)±\sqrt{144+12\times 6}}{2\left(-3\right)}$$

Multiply 12 times 6.

$$x=\frac{-\left(-12\right)±\sqrt{144+72}}{2\left(-3\right)}$$

Add 144 to 72.

$$x=\frac{-\left(-12\right)±\sqrt{216}}{2\left(-3\right)}$$

Take the square root of 216.

$$x=\frac{-\left(-12\right)±6\sqrt{6}}{2\left(-3\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±6\sqrt{6}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{12±6\sqrt{6}}{-6}$$

Now solve the equation x=\frac{12±6\sqrt{6}}{-6} when ± is plus. Add 12 to 6\sqrt{6}.

$$x=\frac{6\sqrt{6}+12}{-6}$$

Divide 12+6\sqrt{6} by -6.

$$x=-\left(\sqrt{6}+2\right)$$

Now solve the equation x=\frac{12±6\sqrt{6}}{-6} when ± is minus. Subtract 6\sqrt{6} from 12.

$$x=\frac{12-6\sqrt{6}}{-6}$$

Divide 12-6\sqrt{6} by -6.

$$x=\sqrt{6}-2$$

The equation is now solved.

$$x=-\left(\sqrt{6}+2\right) x=\sqrt{6}-2$$

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^{2}, the least common multiple of x^{2},x.

$$6-x\times 12=3x^{2}$$

Subtract 3x^{2} from both sides.

$$6-x\times 12-3x^{2}=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-x\times 12-3x^{2}=-6$$

Multiply -1 and 12 to get -12.

$$-12x-3x^{2}=-6$$

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.

$$-3x^{2}-12x=-6$$

Divide both sides by -3.

$$\frac{-3x^{2}-12x}{-3}=-\frac{6}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{12}{-3}\right)x=-\frac{6}{-3}$$

Divide -12 by -3.

$$x^{2}+4x=-\frac{6}{-3}$$

Divide -6 by -3.

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

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.

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

Square 2.

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

Add 2 to 4.

$$x^{2}+4x+4=6$$

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}.

$$\left(x+2\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{6}$$

Simplify.

$$x+2=\sqrt{6} x+2=-\sqrt{6}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$

Solve for x

$$x=\sqrt{6}-2\approx 0.449489743$$

$$x=-\sqrt{6}-2\approx -4.449489743$$

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^{2}, the least common multiple of x^{2},x.

$$6-x\times 12=3x^{2}$$

Subtract 3x^{2} from both sides.

$$6-x\times 12-3x^{2}=0$$

Multiply -1 and 12 to get -12.

$$6-12x-3x^{2}=0$$

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.

$$-3x^{2}-12x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -12 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-12\right)±\sqrt{144+12\times 6}}{2\left(-3\right)}$$

Multiply 12 times 6.

$$x=\frac{-\left(-12\right)±\sqrt{144+72}}{2\left(-3\right)}$$

Add 144 to 72.

$$x=\frac{-\left(-12\right)±\sqrt{216}}{2\left(-3\right)}$$

Take the square root of 216.

$$x=\frac{-\left(-12\right)±6\sqrt{6}}{2\left(-3\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±6\sqrt{6}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{12±6\sqrt{6}}{-6}$$

Now solve the equation x=\frac{12±6\sqrt{6}}{-6} when ± is plus. Add 12 to 6\sqrt{6}.

$$x=\frac{6\sqrt{6}+12}{-6}$$

Divide 12+6\sqrt{6} by -6.

$$x=-\left(\sqrt{6}+2\right)$$

Now solve the equation x=\frac{12±6\sqrt{6}}{-6} when ± is minus. Subtract 6\sqrt{6} from 12.

$$x=\frac{12-6\sqrt{6}}{-6}$$

Divide 12-6\sqrt{6} by -6.

$$x=\sqrt{6}-2$$

The equation is now solved.

$$x=-\left(\sqrt{6}+2\right) x=\sqrt{6}-2$$

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^{2}, the least common multiple of x^{2},x.

$$6-x\times 12=3x^{2}$$

Subtract 3x^{2} from both sides.

$$6-x\times 12-3x^{2}=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-x\times 12-3x^{2}=-6$$

Multiply -1 and 12 to get -12.

$$-12x-3x^{2}=-6$$

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.

$$-3x^{2}-12x=-6$$

Divide both sides by -3.

$$\frac{-3x^{2}-12x}{-3}=-\frac{6}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{12}{-3}\right)x=-\frac{6}{-3}$$

Divide -12 by -3.

$$x^{2}+4x=-\frac{6}{-3}$$

Divide -6 by -3.

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

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.

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

Square 2.

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

Add 2 to 4.

$$x^{2}+4x+4=6$$

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}.

$$\left(x+2\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{6}$$

Simplify.

$$x+2=\sqrt{6} x+2=-\sqrt{6}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$