Solve 16-16/x+4/x^2=18x^2-54/x^2

$$16 - \frac { 16 } { x } + \frac { 4 } { x ^ { 2 } } = \frac { 18 x ^ { 2 } - 54 } { x ^ { 2 } }$$

Solve for x

$$x=3\sqrt{5}-4\approx 2.708203932$$

$$x=-3\sqrt{5}-4\approx -10.708203932$$

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,x^{2}.

$$x^{2}\times 16-x\times 16+4=18x^{2}-54$$

Subtract 18x^{2} from both sides.

$$x^{2}\times 16-x\times 16+4-18x^{2}=-54$$

Combine x^{2}\times 16 and -18x^{2} to get -2x^{2}.

$$-2x^{2}-x\times 16+4=-54$$

Add 54 to both sides.

$$-2x^{2}-x\times 16+4+54=0$$

Add 4 and 54 to get 58.

$$-2x^{2}-x\times 16+58=0$$

Multiply -1 and 16 to get -16.

$$-2x^{2}-16x+58=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -16 for b, and 58 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-2\right)\times 58}}{2\left(-2\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-2\right)\times 58}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-16\right)±\sqrt{256+8\times 58}}{2\left(-2\right)}$$

Multiply 8 times 58.

$$x=\frac{-\left(-16\right)±\sqrt{256+464}}{2\left(-2\right)}$$

Add 256 to 464.

$$x=\frac{-\left(-16\right)±\sqrt{720}}{2\left(-2\right)}$$

Take the square root of 720.

$$x=\frac{-\left(-16\right)±12\sqrt{5}}{2\left(-2\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±12\sqrt{5}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{16±12\sqrt{5}}{-4}$$

Now solve the equation x=\frac{16±12\sqrt{5}}{-4} when ± is plus. Add 16 to 12\sqrt{5}.

$$x=\frac{12\sqrt{5}+16}{-4}$$

Divide 16+12\sqrt{5} by -4.

$$x=-3\sqrt{5}-4$$

Now solve the equation x=\frac{16±12\sqrt{5}}{-4} when ± is minus. Subtract 12\sqrt{5} from 16.

$$x=\frac{16-12\sqrt{5}}{-4}$$

Divide 16-12\sqrt{5} by -4.

$$x=3\sqrt{5}-4$$

The equation is now solved.

$$x=-3\sqrt{5}-4 x=3\sqrt{5}-4$$

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,x^{2}.

$$x^{2}\times 16-x\times 16+4=18x^{2}-54$$

Subtract 18x^{2} from both sides.

$$x^{2}\times 16-x\times 16+4-18x^{2}=-54$$

Combine x^{2}\times 16 and -18x^{2} to get -2x^{2}.

$$-2x^{2}-x\times 16+4=-54$$

Subtract 4 from both sides.

$$-2x^{2}-x\times 16=-54-4$$

Subtract 4 from -54 to get -58.

$$-2x^{2}-x\times 16=-58$$

Multiply -1 and 16 to get -16.

$$-2x^{2}-16x=-58$$

Divide both sides by -2.

$$\frac{-2x^{2}-16x}{-2}=-\frac{58}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{16}{-2}\right)x=-\frac{58}{-2}$$

Divide -16 by -2.

$$x^{2}+8x=-\frac{58}{-2}$$

Divide -58 by -2.

$$x^{2}+8x=29$$

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.

$$x^{2}+8x+4^{2}=29+4^{2}$$

Square 4.

$$x^{2}+8x+16=29+16$$

Add 29 to 16.

$$x^{2}+8x+16=45$$

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}.

$$\left(x+4\right)^{2}=45$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{45}$$

Simplify.

$$x+4=3\sqrt{5} x+4=-3\sqrt{5}$$

Subtract 4 from both sides of the equation.

$$x=3\sqrt{5}-4 x=-3\sqrt{5}-4$$