Solve s+100/s=20

$$s + \frac { 100 } { s } = 20$$

Solve for s

$$s=10$$

Steps Using Factoring

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

$$ss+100=20s$$

Multiply s and s to get s^{2}.

$$s^{2}+100=20s$$

Subtract 20s from both sides.

$$s^{2}+100-20s=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$s^{2}-20s+100=0$$

To solve the equation, factor s^{2}-20s+100 using formula s^{2}+\left(a+b\right)s+ab=\left(s+a\right)\left(s+b\right). To find a and b, set up a system to be solved.

$$a+b=-20 ab=100$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Calculate the sum for each pair.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

The solution is the pair that gives sum -20.

$$a=-10 b=-10$$

Rewrite factored expression \left(s+a\right)\left(s+b\right) using the obtained values.

$$\left(s-10\right)\left(s-10\right)$$

Rewrite as a binomial square.

$$\left(s-10\right)^{2}$$

To find equation solution, solve s-10=0.

$$s=10$$

Steps Using Factoring By Grouping

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

$$ss+100=20s$$

Multiply s and s to get s^{2}.

$$s^{2}+100=20s$$

Subtract 20s from both sides.

$$s^{2}+100-20s=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$s^{2}-20s+100=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as s^{2}+as+bs+100. To find a and b, set up a system to be solved.

$$a+b=-20 ab=1\times 100=100$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Calculate the sum for each pair.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

The solution is the pair that gives sum -20.

$$a=-10 b=-10$$

Rewrite s^{2}-20s+100 as \left(s^{2}-10s\right)+\left(-10s+100\right).

$$\left(s^{2}-10s\right)+\left(-10s+100\right)$$

Factor out s in the first and -10 in the second group.

$$s\left(s-10\right)-10\left(s-10\right)$$

Factor out common term s-10 by using distributive property.

$$\left(s-10\right)\left(s-10\right)$$

Rewrite as a binomial square.

$$\left(s-10\right)^{2}$$

To find equation solution, solve s-10=0.

$$s=10$$

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.

$$ss+100=20s$$

Multiply s and s to get s^{2}.

$$s^{2}+100=20s$$

Subtract 20s from both sides.

$$s^{2}+100-20s=0$$

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.

$$s^{2}-20s+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 100}}{2}$$

Square -20.

$$s=\frac{-\left(-20\right)±\sqrt{400-4\times 100}}{2}$$

Multiply -4 times 100.

$$s=\frac{-\left(-20\right)±\sqrt{400-400}}{2}$$

Add 400 to -400.

$$s=\frac{-\left(-20\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$s=-\frac{-20}{2}$$

The opposite of -20 is 20.

$$s=\frac{20}{2}$$

Divide 20 by 2.

$$s=10$$

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.

$$ss+100=20s$$

Multiply s and s to get s^{2}.

$$s^{2}+100=20s$$

Subtract 20s from both sides.

$$s^{2}+100-20s=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$s^{2}-20s=-100$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}-20s+\left(-10\right)^{2}=-100+\left(-10\right)^{2}$$

Square -10.

$$s^{2}-20s+100=-100+100$$

Add -100 to 100.

$$s^{2}-20s+100=0$$

Factor s^{2}-20s+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(s-10\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s-10\right)^{2}}=\sqrt{0}$$

Simplify.

$$s-10=0 s-10=0$$

Add 10 to both sides of the equation.

$$s=10 s=10$$

The equation is now solved. Solutions are the same.

$$s=10$$