Solve 3/x=x-2/4

$$\frac { 3 } { x } = \frac { x - 2 } { 4 }$$

Solve for x

$$x=\sqrt{13}+1\approx 4.605551275$$

$$x=1-\sqrt{13}\approx -2.605551275$$

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 x,4.

$$4\times 3=x\left(x-2\right)$$

Multiply 4 and 3 to get 12.

$$12=x\left(x-2\right)$$

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

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 12 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -2.

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

Multiply -4 times -12.

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

Add 4 to 48.

$$x=\frac{-\left(-2\right)±\sqrt{52}}{2}$$

Take the square root of 52.

$$x=\frac{-\left(-2\right)±2\sqrt{13}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{13}}{2}$$

Now solve the equation x=\frac{2±2\sqrt{13}}{2} when ± is plus. Add 2 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}+2}{2}$$

Divide 2+2\sqrt{13} by 2.

$$x=\sqrt{13}+1$$

Now solve the equation x=\frac{2±2\sqrt{13}}{2} when ± is minus. Subtract 2\sqrt{13} from 2.

$$x=\frac{2-2\sqrt{13}}{2}$$

Divide 2-2\sqrt{13} by 2.

$$x=1-\sqrt{13}$$

The equation is now solved.

$$x=\sqrt{13}+1 x=1-\sqrt{13}$$

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 x,4.

$$4\times 3=x\left(x-2\right)$$

Multiply 4 and 3 to get 12.

$$12=x\left(x-2\right)$$

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

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

Swap sides so that all variable terms are on the left hand side.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=12+1$$

Add 12 to 1.

$$x^{2}-2x+1=13$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{13}$$

Simplify.

$$x-1=\sqrt{13} x-1=-\sqrt{13}$$

Add 1 to both sides of the equation.

$$x=\sqrt{13}+1 x=1-\sqrt{13}$$