Solve s/s+2+s=5s+8/s+2

$$\frac { s } { s + 2 } + s = \frac { 5 s + 8 } { s + 2 }$$

Solve for s

$$s=4$$

Steps Using the Quadratic Formula

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

$$s+\left(s+2\right)s=5s+8$$

Use the distributive property to multiply s+2 by s.

$$s+s^{2}+2s=5s+8$$

Combine s and 2s to get 3s.

$$3s+s^{2}=5s+8$$

Subtract 5s from both sides.

$$3s+s^{2}-5s=8$$

Combine 3s and -5s to get -2s.

$$-2s+s^{2}=8$$

Subtract 8 from both sides.

$$-2s+s^{2}-8=0$$

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.

$$s^{2}-2s-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-8\right)}}{2}$$

Square -2.

$$s=\frac{-\left(-2\right)±\sqrt{4-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$s=\frac{-\left(-2\right)±\sqrt{4+32}}{2}$$

Add 4 to 32.

$$s=\frac{-\left(-2\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$s=\frac{-\left(-2\right)±6}{2}$$

The opposite of -2 is 2.

$$s=\frac{2±6}{2}$$

Now solve the equation s=\frac{2±6}{2} when ± is plus. Add 2 to 6.

$$s=\frac{8}{2}$$

Divide 8 by 2.

$$s=4$$

Now solve the equation s=\frac{2±6}{2} when ± is minus. Subtract 6 from 2.

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

Divide -4 by 2.

$$s=-2$$

The equation is now solved.

$$s=4 s=-2$$

Variable s cannot be equal to -2.

$$s=4$$

Steps for Completing the Square

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

$$s+\left(s+2\right)s=5s+8$$

Use the distributive property to multiply s+2 by s.

$$s+s^{2}+2s=5s+8$$

Combine s and 2s to get 3s.

$$3s+s^{2}=5s+8$$

Subtract 5s from both sides.

$$3s+s^{2}-5s=8$$

Combine 3s and -5s to get -2s.

$$-2s+s^{2}=8$$

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.

$$s^{2}-2s=8$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}-2s+1=8+1$$

Add 8 to 1.

$$s^{2}-2s+1=9$$

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

$$\left(s-1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s-1\right)^{2}}=\sqrt{9}$$

Simplify.

$$s-1=3 s-1=-3$$

Add 1 to both sides of the equation.

$$s=4 s=-2$$

Variable s cannot be equal to -2.

$$s=4$$