## Geometry: Common Core (15th Edition)

$x = 3$ or $x = \frac{2}{7}$
According to the side-splitter theorem, if a line is parallel to a side of a triangle and intersects the other two sides, then those two sides are divided proportionately. Let's set up a proportion for the sides whose values are given in the figure: $\frac{12}{7x} = \frac{2x + 2}{5x - 1}$ Use the cross product property to get rid of the fractions: $7x(2x + 2) = 12(5x - 1)$ Use the distributive property first: $14x^2 + 14x = 60x - 12$ Move all terms to the left side of the equation: $14x^2 + 14x - 60x + 12 = 0$ Combine like terms: $14x^2 - 46x + 12 = 0$ Factor out a $2$ from all terms: $7x^2 - 23x + 6 = 0$ Use the quadratic formula to solve this equation: $x = \frac{-(-23) ± \sqrt {(-23)^2 - 4(7)(6)}}{2(7)}$ Simplify by multiplying: $x = \frac{23 ± \sqrt {529 - 168}}{14}$ Simplify what is under the square root sign: $x = \frac{23 ± \sqrt {361}}{14}$ Take the square root of $361$: $x = \frac{23 ± 19}{14}$ Add or subtract to simplify the fraction: $x = \frac{42}{14}$ or $x = \frac{4}{14}$ Simplify the fractions by dividing their numerators and denominators by their greatest common factors: $x = 3$ or $x = \frac{2}{7}$