Solve 2x+4/x-2=2x+1

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

Solve for x

$$x = \frac{\sqrt{73} + 5}{4} \approx 3.386000936$$

$$x=\frac{5-\sqrt{73}}{4}\approx -0.886000936$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

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

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

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

Combine -4x and x to get -3x.

$$2x+4=2x^{2}-3x-2$$

Subtract 2x^{2} from both sides.

$$2x+4-2x^{2}=-3x-2$$

Add 3x to both sides.

$$2x+4-2x^{2}+3x=-2$$

Combine 2x and 3x to get 5x.

$$5x+4-2x^{2}=-2$$

Add 2 to both sides.

$$5x+4-2x^{2}+2=0$$

Add 4 and 2 to get 6.

$$5x+6-2x^{2}=0$$

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.

$$-2x^{2}+5x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 5 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-5±\sqrt{25+8\times 6}}{2\left(-2\right)}$$

Multiply 8 times 6.

$$x=\frac{-5±\sqrt{25+48}}{2\left(-2\right)}$$

Add 25 to 48.

$$x=\frac{-5±\sqrt{73}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-5±\sqrt{73}}{-4}$$

Now solve the equation x=\frac{-5±\sqrt{73}}{-4} when ± is plus. Add -5 to \sqrt{73}.

$$x=\frac{\sqrt{73}-5}{-4}$$

Divide -5+\sqrt{73} by -4.

$$x=\frac{5-\sqrt{73}}{4}$$

Now solve the equation x=\frac{-5±\sqrt{73}}{-4} when ± is minus. Subtract \sqrt{73} from -5.

$$x=\frac{-\sqrt{73}-5}{-4}$$

Divide -5-\sqrt{73} by -4.

$$x=\frac{\sqrt{73}+5}{4}$$

The equation is now solved.

$$x=\frac{5-\sqrt{73}}{4} x=\frac{\sqrt{73}+5}{4}$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

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

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

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

Combine -4x and x to get -3x.

$$2x+4=2x^{2}-3x-2$$

Subtract 2x^{2} from both sides.

$$2x+4-2x^{2}=-3x-2$$

Add 3x to both sides.

$$2x+4-2x^{2}+3x=-2$$

Combine 2x and 3x to get 5x.

$$5x+4-2x^{2}=-2$$

Subtract 4 from both sides.

$$5x-2x^{2}=-2-4$$

Subtract 4 from -2 to get -6.

$$5x-2x^{2}=-6$$

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.

$$-2x^{2}+5x=-6$$

Divide both sides by -2.

$$\frac{-2x^{2}+5x}{-2}=-\frac{6}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{5}{-2}x=-\frac{6}{-2}$$

Divide 5 by -2.

$$x^{2}-\frac{5}{2}x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x^{2}-\frac{5}{2}x=3$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=3+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=3+\frac{25}{16}$$

Add 3 to \frac{25}{16}.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{73}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{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}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{73}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{73}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{73}}{4} x-\frac{5}{4}=-\frac{\sqrt{73}}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{\sqrt{73}+5}{4} x=\frac{5-\sqrt{73}}{4}$$