Answer
$x = \frac{7}{2}$ (in thousand units)
$p (x) = 65.5$ USD
Work Step by Step
To reach market equilibrium, the demand and supply function must be equal.
Thus, we have: $$-2x^{2} + 90 = 9x + 34$$
Simplify the equation: $$-2x^{2}-9x+56 = 0$$
Using the quadratic formula to get the roots: $$x = \frac{-b ± \sqrt{(b^{2}-4ac)}}{2a}$$ $$x = \frac{-(-9) ± \sqrt{ ((-9)^{2}-4(-2)(56)}}{2(-2)}$$ $$x = \frac{9 ± \sqrt {529}}{-4}$$ $$x = −8$$ $$x= 7/2$$
Since we are talking about the number of units produced, $x$ cannot take a negative value; hence, we take $7/2$.
Substituting to the supply equation: $$p(x) =9x + 34$$ $$p(x) =9(\frac{7}{2}) + 34$$ $$p(x) = 65.5$$
