Solve 2(x)=x^2/x-4x-1

$$2 ( x ) = \frac { x ^ { 2 } } { x } - 4 x - 1$$

Solve for x

$$x=-\frac{1}{5}=-0.2$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$2xx=x^{2}-4xx+x\left(-1\right)$$

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

$$2x^{2}=x^{2}-4xx+x\left(-1\right)$$

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

$$2x^{2}=x^{2}-4x^{2}+x\left(-1\right)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$2x^{2}=-3x^{2}+x\left(-1\right)$$

Add 3x^{2} to both sides.

$$2x^{2}+3x^{2}=x\left(-1\right)$$

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}=x\left(-1\right)$$

Subtract x\left(-1\right) from both sides.

$$5x^{2}-x\left(-1\right)=0$$

Multiply -1 and -1 to get 1.

$$5x^{2}+x=0$$

Factor out x.

$$x\left(5x+1\right)=0$$

To find equation solutions, solve x=0 and 5x+1=0.

$$x=0 x=-\frac{1}{5}$$

Variable x cannot be equal to 0.

$$x=-\frac{1}{5}$$

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.

$$2xx=x^{2}-4xx+x\left(-1\right)$$

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

$$2x^{2}=x^{2}-4xx+x\left(-1\right)$$

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

$$2x^{2}=x^{2}-4x^{2}+x\left(-1\right)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$2x^{2}=-3x^{2}+x\left(-1\right)$$

Add 3x^{2} to both sides.

$$2x^{2}+3x^{2}=x\left(-1\right)$$

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}=x\left(-1\right)$$

Subtract x\left(-1\right) from both sides.

$$5x^{2}-x\left(-1\right)=0$$

Multiply -1 and -1 to get 1.

$$5x^{2}+x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}}}{2\times 5}$$

Take the square root of 1^{2}.

$$x=\frac{-1±1}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-1±1}{10}$$

Now solve the equation x=\frac{-1±1}{10} when ± is plus. Add -1 to 1.

$$x=\frac{0}{10}$$

Divide 0 by 10.

$$x=0$$

Now solve the equation x=\frac{-1±1}{10} when ± is minus. Subtract 1 from -1.

$$x=-\frac{2}{10}$$

Reduce the fraction \frac{-2}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{5}$$

The equation is now solved.

$$x=0 x=-\frac{1}{5}$$

Variable x cannot be equal to 0.

$$x=-\frac{1}{5}$$

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.

$$2xx=x^{2}-4xx+x\left(-1\right)$$

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

$$2x^{2}=x^{2}-4xx+x\left(-1\right)$$

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

$$2x^{2}=x^{2}-4x^{2}+x\left(-1\right)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$2x^{2}=-3x^{2}+x\left(-1\right)$$

Add 3x^{2} to both sides.

$$2x^{2}+3x^{2}=x\left(-1\right)$$

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}=x\left(-1\right)$$

Subtract x\left(-1\right) from both sides.

$$5x^{2}-x\left(-1\right)=0$$

Multiply -1 and -1 to get 1.

$$5x^{2}+x=0$$

Divide both sides by 5.

$$\frac{5x^{2}+x}{5}=\frac{0}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{1}{5}x=\frac{0}{5}$$

Divide 0 by 5.

$$x^{2}+\frac{1}{5}x=0$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{1}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{1}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{1}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{1}{10} x+\frac{1}{10}=-\frac{1}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=0 x=-\frac{1}{5}$$

Variable x cannot be equal to 0.

$$x=-\frac{1}{5}$$