Solve (S-2)*80=2(360/S)

$$( S - 2 ) \cdot 80 = 2 ( \frac { 360 } { S } )$$

Solve for S

$$S=\sqrt{10}+1\approx 4.16227766$$

$$S=1-\sqrt{10}\approx -2.16227766$$

Steps Using the Quadratic Formula

Variable S cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by S.

$$\left(S-2\right)\times 80S=2\times 360$$

Use the distributive property to multiply S-2 by 80.

$$\left(80S-160\right)S=2\times 360$$

Use the distributive property to multiply 80S-160 by S.

$$80S^{2}-160S=2\times 360$$

Multiply 2 and 360 to get 720.

$$80S^{2}-160S=720$$

Subtract 720 from both sides.

$$80S^{2}-160S-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 80 for a, -160 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$S=\frac{-\left(-160\right)±\sqrt{\left(-160\right)^{2}-4\times 80\left(-720\right)}}{2\times 80}$$

Square -160.

$$S=\frac{-\left(-160\right)±\sqrt{25600-4\times 80\left(-720\right)}}{2\times 80}$$

Multiply -4 times 80.

$$S=\frac{-\left(-160\right)±\sqrt{25600-320\left(-720\right)}}{2\times 80}$$

Multiply -320 times -720.

$$S=\frac{-\left(-160\right)±\sqrt{25600+230400}}{2\times 80}$$

Add 25600 to 230400.

$$S=\frac{-\left(-160\right)±\sqrt{256000}}{2\times 80}$$

Take the square root of 256000.

$$S=\frac{-\left(-160\right)±160\sqrt{10}}{2\times 80}$$

The opposite of -160 is 160.

$$S=\frac{160±160\sqrt{10}}{2\times 80}$$

Multiply 2 times 80.

$$S=\frac{160±160\sqrt{10}}{160}$$

Now solve the equation S=\frac{160±160\sqrt{10}}{160} when ± is plus. Add 160 to 160\sqrt{10}.

$$S=\frac{160\sqrt{10}+160}{160}$$

Divide 160+160\sqrt{10} by 160.

$$S=\sqrt{10}+1$$

Now solve the equation S=\frac{160±160\sqrt{10}}{160} when ± is minus. Subtract 160\sqrt{10} from 160.

$$S=\frac{160-160\sqrt{10}}{160}$$

Divide 160-160\sqrt{10} by 160.

$$S=1-\sqrt{10}$$

The equation is now solved.

$$S=\sqrt{10}+1 S=1-\sqrt{10}$$

Steps for Completing the Square

Variable S cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by S.

$$\left(S-2\right)\times 80S=2\times 360$$

Use the distributive property to multiply S-2 by 80.

$$\left(80S-160\right)S=2\times 360$$

Use the distributive property to multiply 80S-160 by S.

$$80S^{2}-160S=2\times 360$$

Multiply 2 and 360 to get 720.

$$80S^{2}-160S=720$$

Divide both sides by 80.

$$\frac{80S^{2}-160S}{80}=\frac{720}{80}$$

Dividing by 80 undoes the multiplication by 80.

$$S^{2}+\left(-\frac{160}{80}\right)S=\frac{720}{80}$$

Divide -160 by 80.

$$S^{2}-2S=\frac{720}{80}$$

Divide 720 by 80.

$$S^{2}-2S=9$$

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.

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

Add 9 to 1.

$$S^{2}-2S+1=10$$

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}.

$$\left(S-1\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(S-1\right)^{2}}=\sqrt{10}$$

Simplify.

$$S-1=\sqrt{10} S-1=-\sqrt{10}$$

Add 1 to both sides of the equation.

$$S=\sqrt{10}+1 S=1-\sqrt{10}$$