```Question 104060
he demand and supply equations for a certain item are given by
D = –5p + 40
S = –p^2 + 30p – 8
Find the equilibrium price.
:
The equilibrium price (p) occurs when D = S, therefore:
:
-5p + 40 = -p^2 + 30p - 8
:
Arrange as a quadratic equation on the left:
+p^2 - 5p - 30p + 40 + 8 = 0
:
p^2 - 35p + 48 = 0
:
This equation requires the quadratic formula to solve: a = 1; b =-35; c= 48
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
:
{{{p = (-(-35) +- sqrt(-35^2 - 4 * 1 * 48 ))/(2*1) }}}
:
{{{p = (35 +- sqrt(1225 - 192 ))/(2) }}}
:
{{{p = (35 +- sqrt(1033 ))/(2) }}}
:
{{{p = (35 +- 32.14)/(2) }}}
Two solutions
{{{p = (35 + 32.14)/(2) }}}
{{{p = (67.14)/(2) }}}
p = 33.57
and
{{{p = (35 - 32.14)/(2) }}}
{{{p = (2.86)/(2) }}}
p = \$1.43
:
Only the lower price make sense, the higher value will give negative values in the original expressions.
:
You can check this by substituting 1.43 for p
-5(1.43) + 40 = -(1.43^2) + 30p - 8
-7.15 + 40 = -2.0449 + 30(1.43) - 8
32.85 = -2.0449 + 42.9 - 8
32.85 = 32.85 equality reigns
:
Did this make sense to you? Did it help? Any question?

```