Solve -2x/x+4=2x+5

$$\frac{ -2x }{ x+4 } =2x+5$$

Solve for x

$$x=\frac{\sqrt{65}-15}{4}\approx -1.734435563$$

$$x=\frac{-\sqrt{65}-15}{4}\approx -5.765564437$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$-2x=2x\left(x+4\right)+\left(x+4\right)\times 5$$

Use the distributive property to multiply 2x by x+4.

$$-2x=2x^{2}+8x+\left(x+4\right)\times 5$$

Use the distributive property to multiply x+4 by 5.

$$-2x=2x^{2}+8x+5x+20$$

Combine 8x and 5x to get 13x.

$$-2x=2x^{2}+13x+20$$

Subtract 2x^{2} from both sides.

$$-2x-2x^{2}=13x+20$$

Subtract 13x from both sides.

$$-2x-2x^{2}-13x=20$$

Combine -2x and -13x to get -15x.

$$-15x-2x^{2}=20$$

Subtract 20 from both sides.

$$-15x-2x^{2}-20=0$$

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.

$$-2x^{2}-15x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -15 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-2\right)\left(-20\right)}}{2\left(-2\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-2\right)\left(-20\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-15\right)±\sqrt{225+8\left(-20\right)}}{2\left(-2\right)}$$

Multiply 8 times -20.

$$x=\frac{-\left(-15\right)±\sqrt{225-160}}{2\left(-2\right)}$$

Add 225 to -160.

$$x=\frac{-\left(-15\right)±\sqrt{65}}{2\left(-2\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{65}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{15±\sqrt{65}}{-4}$$

Now solve the equation x=\frac{15±\sqrt{65}}{-4} when ± is plus. Add 15 to \sqrt{65}.

$$x=\frac{\sqrt{65}+15}{-4}$$

Divide 15+\sqrt{65} by -4.

$$x=\frac{-\sqrt{65}-15}{4}$$

Now solve the equation x=\frac{15±\sqrt{65}}{-4} when ± is minus. Subtract \sqrt{65} from 15.

$$x=\frac{15-\sqrt{65}}{-4}$$

Divide 15-\sqrt{65} by -4.

$$x=\frac{\sqrt{65}-15}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{65}-15}{4} x=\frac{\sqrt{65}-15}{4}$$

Steps for Completing the Square

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$-2x=2x\left(x+4\right)+\left(x+4\right)\times 5$$

Use the distributive property to multiply 2x by x+4.

$$-2x=2x^{2}+8x+\left(x+4\right)\times 5$$

Use the distributive property to multiply x+4 by 5.

$$-2x=2x^{2}+8x+5x+20$$

Combine 8x and 5x to get 13x.

$$-2x=2x^{2}+13x+20$$

Subtract 2x^{2} from both sides.

$$-2x-2x^{2}=13x+20$$

Subtract 13x from both sides.

$$-2x-2x^{2}-13x=20$$

Combine -2x and -13x to get -15x.

$$-15x-2x^{2}=20$$

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.

$$-2x^{2}-15x=20$$

Divide both sides by -2.

$$\frac{-2x^{2}-15x}{-2}=\frac{20}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{15}{-2}\right)x=\frac{20}{-2}$$

Divide -15 by -2.

$$x^{2}+\frac{15}{2}x=\frac{20}{-2}$$

Divide 20 by -2.

$$x^{2}+\frac{15}{2}x=-10$$

Divide \frac{15}{2}, the coefficient of the x term, by 2 to get \frac{15}{4}. Then add the square of \frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{2}x+\left(\frac{15}{4}\right)^{2}=-10+\left(\frac{15}{4}\right)^{2}$$

Square \frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=-10+\frac{225}{16}$$

Add -10 to \frac{225}{16}.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=\frac{65}{16}$$

Factor x^{2}+\frac{15}{2}x+\frac{225}{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}.

$$\left(x+\frac{15}{4}\right)^{2}=\frac{65}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{4}\right)^{2}}=\sqrt{\frac{65}{16}}$$

Simplify.

$$x+\frac{15}{4}=\frac{\sqrt{65}}{4} x+\frac{15}{4}=-\frac{\sqrt{65}}{4}$$

Subtract \frac{15}{4} from both sides of the equation.

$$x=\frac{\sqrt{65}-15}{4} x=\frac{-\sqrt{65}-15}{4}$$