Solve 4x-1/x+1=x-1

$$\frac { 4 x - 1 } { x + 1 } = x - 1$$

Solve for x

$$x=4$$

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$4x-1=\left(x+1\right)x+\left(x+1\right)\left(-1\right)$$

Use the distributive property to multiply x+1 by x.

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

Use the distributive property to multiply x+1 by -1.

$$4x-1=x^{2}+x-x-1$$

Combine x and -x to get 0.

$$4x-1=x^{2}-1$$

Subtract x^{2} from both sides.

$$4x-1-x^{2}=-1$$

Add 1 to both sides.

$$4x-1-x^{2}+1=0$$

Add -1 and 1 to get 0.

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

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.

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

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

$$x=\frac{-4±\sqrt{4^{2}}}{2\left(-1\right)}$$

Take the square root of 4^{2}.

$$x=\frac{-4±4}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±4}{-2}$$

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

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

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

$$x=-\frac{8}{-2}$$

Divide -8 by -2.

$$x=4$$

The equation is now solved.

$$x=0 x=4$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$4x-1=\left(x+1\right)x+\left(x+1\right)\left(-1\right)$$

Use the distributive property to multiply x+1 by x.

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

Use the distributive property to multiply x+1 by -1.

$$4x-1=x^{2}+x-x-1$$

Combine x and -x to get 0.

$$4x-1=x^{2}-1$$

Subtract x^{2} from both sides.

$$4x-1-x^{2}=-1$$

Add 1 to both sides.

$$4x-x^{2}=-1+1$$

Add -1 and 1 to get 0.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{0}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{0}{-1}$$

Divide 0 by -1.

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

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.

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

Square -2.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

$$x=4 x=0$$