Answer
$x = 15\:units$
$p(x) = 3.75\:USD$
Work Step by Step
To reach market equilibrium, the demand and supply function must be equal.
Thus, we have: $$-0.01x^{2} -0.2x + 9 = 0.01x^{2} -0.1x + 3$$
Simplify the equation: $$-0.02x^{2}-0.1x+6 = 0$$
Using the quadratic formula to get the roots: $$x = \frac{-b ± \sqrt{(b^{2}-4ac)}}{2a}$$ $$x = \frac{-(-0.1) ± \sqrt{((-0.1)^{2}-4(-0.02)(6)}}{2(-0.02)}$$ $$x = \frac{0.1 ± \sqrt {0.49}}{-0.04}$$ $$x = −19.999999999999996$$ $$x= 15$$
Since we are talking about the number of units produced, $x$ cannot take a negative value; hence, we take $15$.
Substituting to the demand equation: $$p(x) =-0.01x^{2} -0.2x + 9$$ $$p(x) =-0.01(15)^{2} -0.2(15) + 9$$ $$p(x) =-2.25 - 3 + 9$$ $$p(x) = 3.75$$
