Solve 12/x=x-4

$$\frac { 12 } { x } = x - 4$$

Solve for x

$$x=-2$$

$$x=6$$

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.

$$12=xx+x\left(-4\right)$$

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

$$12=x^{2}+x\left(-4\right)$$

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

$$x^{2}+x\left(-4\right)=12$$

Subtract 12 from both sides.

$$x^{2}+x\left(-4\right)-12=0$$

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.

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

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

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-12\right)}}{2}$$

Multiply -4 times -12.

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

Add 16 to 48.

$$x=\frac{-\left(-4\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-\left(-4\right)±8}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±8}{2}$$

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

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

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

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

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=6 x=-2$$

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.

$$12=xx+x\left(-4\right)$$

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

$$12=x^{2}+x\left(-4\right)$$

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

$$x^{2}+x\left(-4\right)=12$$

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.

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

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.

$$x^{2}-4x+\left(-2\right)^{2}=12+\left(-2\right)^{2}$$

Square -2.

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

Add 12 to 4.

$$x^{2}-4x+4=16$$

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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-2=4 x-2=-4$$

Add 2 to both sides of the equation.

$$x=6 x=-2$$