Solve x^2/99+20x/99+1=0

$$\frac { x ^ { 2 } } { 99 } + \frac { 20 x } { 99 } + 1 = 0$$

Solve for x

$$x=-11$$

$$x=-9$$

Steps Using Factoring

Multiply both sides of the equation by 99.

$$x^{2}+20x+99=0$$

To solve the equation, factor x^{2}+20x+99 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=20 ab=99$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 99.

$$1,99 3,33 9,11$$

Calculate the sum for each pair.

$$1+99=100 3+33=36 9+11=20$$

The solution is the pair that gives sum 20.

$$a=9 b=11$$

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

$$\left(x+9\right)\left(x+11\right)$$

To find equation solutions, solve x+9=0 and x+11=0.

$$x=-9 x=-11$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 99.

$$x^{2}+20x+99=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+99. To find a and b, set up a system to be solved.

$$a+b=20 ab=1\times 99=99$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 99.

$$1,99 3,33 9,11$$

Calculate the sum for each pair.

$$1+99=100 3+33=36 9+11=20$$

The solution is the pair that gives sum 20.

$$a=9 b=11$$

Rewrite x^{2}+20x+99 as \left(x^{2}+9x\right)+\left(11x+99\right).

$$\left(x^{2}+9x\right)+\left(11x+99\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x+9\right)+11\left(x+9\right)$$

Factor out common term x+9 by using distributive property.

$$\left(x+9\right)\left(x+11\right)$$

To find equation solutions, solve x+9=0 and x+11=0.

$$x=-9 x=-11$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 99.

$$x^{2}+20x+99=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and 99 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 99}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 99}}{2}$$

Multiply -4 times 99.

$$x=\frac{-20±\sqrt{400-396}}{2}$$

Add 400 to -396.

$$x=\frac{-20±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-20±2}{2}$$

Now solve the equation x=\frac{-20±2}{2} when ± is plus. Add -20 to 2.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

Now solve the equation x=\frac{-20±2}{2} when ± is minus. Subtract 2 from -20.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=-9 x=-11$$

Steps for Completing the Square

Multiply both sides of the equation by 99.

$$x^{2}+20x+99=0$$

Subtract 99 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+20x=-99$$

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.

$$x^{2}+20x+10^{2}=-99+10^{2}$$

Square 10.

$$x^{2}+20x+100=-99+100$$

Add -99 to 100.

$$x^{2}+20x+100=1$$

Factor x^{2}+20x+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}.

$$\left(x+10\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+10=1 x+10=-1$$

Subtract 10 from both sides of the equation.

$$x=-9 x=-11$$