Answer
$x = 3$ or $x = \frac{2}{7}$
Work Step by Step
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 $x = \frac{2}{7}$
