Answer:
equilibrium point (10,340)
Step-by-step explanation:
To find the equilibrium point, equal the demand and the supply:
![D(q)=S(q)\\\\-6.10q^2-5q+1000=3.2q^2+10q-80](https://tex.z-dn.net/?f=D%28q%29%3DS%28q%29%5C%5C%5C%5C-6.10q%5E2-5q%2B1000%3D3.2q%5E2%2B10q-80)
Reorganize the terms in one side and reduce similar terms:
![3.2q^2+6.1q^2+5q+10q-80-1000=0\\\\9.3q^2+15q-1080=0](https://tex.z-dn.net/?f=3.2q%5E2%2B6.1q%5E2%2B5q%2B10q-80-1000%3D0%5C%5C%5C%5C9.3q%5E2%2B15q-1080%3D0)
that's a cuadratic equation, solve with the general formula when:
<u><em>a=9.3</em></u>, <u><em>b=15</em></u>, <u><em>c=-1080</em></u>
![q_{1}=\frac{-b+\sqrt{b^{2}-4ac} }{2a}\\\\q_{2}=\frac{-b-\sqrt{b^{2}-4ac} }{2a}\\\\q_{1}=\frac{-15+\sqrt{(-15)^{2}-4(9.3)(-1080)} }{2(9.3)}\\\\q_{1}=\frac{-15+201}{18.6}\\\\q_{1}=\frac{186}{18.6}\\\\q_1=10](https://tex.z-dn.net/?f=q_%7B1%7D%3D%5Cfrac%7B-b%2B%5Csqrt%7Bb%5E%7B2%7D-4ac%7D%20%7D%7B2a%7D%5C%5C%5C%5Cq_%7B2%7D%3D%5Cfrac%7B-b-%5Csqrt%7Bb%5E%7B2%7D-4ac%7D%20%7D%7B2a%7D%5C%5C%5C%5Cq_%7B1%7D%3D%5Cfrac%7B-15%2B%5Csqrt%7B%28-15%29%5E%7B2%7D-4%289.3%29%28-1080%29%7D%20%7D%7B2%289.3%29%7D%5C%5C%5C%5Cq_%7B1%7D%3D%5Cfrac%7B-15%2B201%7D%7B18.6%7D%5C%5C%5C%5Cq_%7B1%7D%3D%5Cfrac%7B186%7D%7B18.6%7D%5C%5C%5C%5Cq_1%3D10)
<em><u>q</u></em> can't be negative because it is the quantity of bike frames, so:
![q_{2}=\frac{-b-\sqrt{b^{2}-4ac} }{2a}\\\\q_{2}=\frac{-15-\sqrt{(-15)^{2}-4(9.3)(-1080)} }{2(9.3)}\\\\q_{2}=\frac{-15-201}{18.6}\\\\q_{2}=\frac{-216}{18.6}\\\\](https://tex.z-dn.net/?f=q_%7B2%7D%3D%5Cfrac%7B-b-%5Csqrt%7Bb%5E%7B2%7D-4ac%7D%20%7D%7B2a%7D%5C%5C%5C%5Cq_%7B2%7D%3D%5Cfrac%7B-15-%5Csqrt%7B%28-15%29%5E%7B2%7D-4%289.3%29%28-1080%29%7D%20%7D%7B2%289.3%29%7D%5C%5C%5C%5Cq_%7B2%7D%3D%5Cfrac%7B-15-201%7D%7B18.6%7D%5C%5C%5C%5Cq_%7B2%7D%3D%5Cfrac%7B-216%7D%7B18.6%7D%5C%5C%5C%5C)
This value of <u><em>q</em></u> can't be considered.
Then substitute the value of <em><u>q</u></em> in <u><em>D(q)</em></u> to find the price <u><em>p</em></u>:
![D(10) = -6.10(10)^2-5(10) + 1000\\\\D(10)=340=p](https://tex.z-dn.net/?f=D%2810%29%20%3D%20-6.10%2810%29%5E2-5%2810%29%20%2B%201000%5C%5C%5C%5CD%2810%29%3D340%3Dp)
The equilibrium point (q,p) is (10,340).