Alg3.3.3

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$\frac{1}{x+2}=\frac{5x}{9}\,$

First, notice that $x\ne -2\,$ .

Second, multiply both sides by $x+2\,$ .

$1=\frac{5x(x+2)}{9}\,$

Multiply both sides by $9\,$ .

$9=5x(x+2)\,$

Expand the right hand side.

$9=5x^2+10x\,$

Subtract $5x^2+10x\,$ from both sides so that the equation can be factored. Since $x\,$ is raised to the second power, expect 2 answers.

$-5x^2-10x+9=0\,$

Use the quadratic formula to factor:

$ax^2+bx+c=0\,$ gives $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\,$

So in the given problem,

$x = \frac{10 \pm \sqrt{100-4*(-45)}}{-10}\,$

Simplifying under the radical,

$x = \frac{10 \pm \sqrt{280}}{-10}\,$

To furthur simplify the radical, factor $280$ as $23 * 5 * 7$ . So $22$ can be taken out of it as $2$ .

$x = \frac{10 \pm 2\sqrt{70}}{-10}\,$

Seperate this fraction by dividing both parts of the numerator by the denominator.

$x = -1 \pm \frac{2}{-10}\sqrt{70}\,$