Solve S-2/2=14-S/S-2

$$\frac { S - 2 } { 2 } = \frac { 14 - S } { S - 2 }$$

Solve for S

$$S=-4$$

$$S=6$$

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 2\left(S-2\right), the least common multiple of 2,S-2.

$$\left(S-2\right)\left(S-2\right)=2\left(14-S\right)$$

Multiply S-2 and S-2 to get \left(S-2\right)^{2}.

$$\left(S-2\right)^{2}=2\left(14-S\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(S-2\right)^{2}.

$$S^{2}-4S+4=2\left(14-S\right)$$

Use the distributive property to multiply 2 by 14-S.

$$S^{2}-4S+4=28-2S$$

Subtract 28 from both sides.

$$S^{2}-4S+4-28=-2S$$

Subtract 28 from 4 to get -24.

$$S^{2}-4S-24=-2S$$

Add 2S to both sides.

$$S^{2}-4S-24+2S=0$$

Combine -4S and 2S to get -2S.

$$S^{2}-2S-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$S=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-24\right)}}{2}$$

Square -2.

$$S=\frac{-\left(-2\right)±\sqrt{4-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$S=\frac{-\left(-2\right)±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$S=\frac{-\left(-2\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$S=\frac{-\left(-2\right)±10}{2}$$

The opposite of -2 is 2.

$$S=\frac{2±10}{2}$$

Now solve the equation S=\frac{2±10}{2} when ± is plus. Add 2 to 10.

$$S=\frac{12}{2}$$

Divide 12 by 2.

$$S=6$$

Now solve the equation S=\frac{2±10}{2} when ± is minus. Subtract 10 from 2.

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

Divide -8 by 2.

$$S=-4$$

The equation is now solved.

$$S=6 S=-4$$

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 2\left(S-2\right), the least common multiple of 2,S-2.

$$\left(S-2\right)\left(S-2\right)=2\left(14-S\right)$$

Multiply S-2 and S-2 to get \left(S-2\right)^{2}.

$$\left(S-2\right)^{2}=2\left(14-S\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(S-2\right)^{2}.

$$S^{2}-4S+4=2\left(14-S\right)$$

Use the distributive property to multiply 2 by 14-S.

$$S^{2}-4S+4=28-2S$$

Add 2S to both sides.

$$S^{2}-4S+4+2S=28$$

Combine -4S and 2S to get -2S.

$$S^{2}-2S+4=28$$

Subtract 4 from both sides.

$$S^{2}-2S=28-4$$

Subtract 4 from 28 to get 24.

$$S^{2}-2S=24$$

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.

$$S^{2}-2S+1=24+1$$

Add 24 to 1.

$$S^{2}-2S+1=25$$

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}.

$$\left(S-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(S-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$S-1=5 S-1=-5$$

Add 1 to both sides of the equation.

$$S=6 S=-4$$