6.1 - Solving Quadratic Equations

Andy and Susie run a custom T-shirt business. From past experience, they can model their expected profit, in dollars, with the relation P = -x2 + 120x – 2000, where x is the number of T-shirts they sell

Andy wants to sell enough T-shirts to earn $1200

Susie wants to sell just enough T0shrits to break even because she wants to close the business

Break Even means profit = $0, so P = 0 0 = -x2 + 120x – 2000

◦ Factor the equation to find x, the number of T-shirts they need to sell to break even

0 = (-x +100)(x - 20) 0 = -(x – 100)(x – 20) Therefore, the zeros are x = 100 and x =

20, so in order to break even, they need to produce 20 T-shirts.

Reflect

Example #1 The user’s manual for Jon’s model

rocket says that the equation h = -5t2 + 40t models the approximate height, in meters, of the rocket after t seconds. When will Jon’s rocket reach a height of 60m?

We need to find t when h = 60 60 = -5t2 + 40t -5t2 + 40t – 60 = 0 -5(t2 - 8t + 12) = 0 -5(t – 6)(t – 2) So, t = 2s and t = 6s

So, the rocket is 60m above the ground at 2s on its way up, and at 6s

on its way down.

Determine the roots of 6x2 – 11x – 10 = 0 (3x + 2)(2x - 5) = 0 So either 3x + 2 = 0, OR 2x – 5 = 0 3x + 2 = 0 3x = -2 x = 2x – 5 = 0 2x = 5 x = Therefore, the roots of the equation are and

Example #2

Determine all the values of x that satisfy the equation x2 + 4 = 3x(x – 5). If necessary, round your answer to 2 decimal places.

x2 + 4 = 3x(x – 5)x2 + 4 = 3x2 – 15x0 = 3x2 – x2 – 15x - 40 = 2x2 – 15x - 4This equation does not have whole number solutions, so to find the x values that satisfy this equation, graph it.

Example #3

0 = 2x2 – 15x - 4 This is the top-half of the parabola We can see that its x-intercepts are at:

◦ x = -0.26 and x = 7.76

Example #3 Cont’d

A ball is thrown from the top of a seaside cliff. Its height, h, in meters, above the sea after t seconds can be modeled by h = -5t2 + 21t + 120. How long will the ball take to fall 20m below its initial height?

1st, we need to know it’s initial height (t=0) h = -5(0)2 + 21(0) + 120 = 120m

Example #4

We are looking for time it takes to fall to 100m

100 = -5t2 + 21t + 120 0 = -5t2 + 21t + 20 0 = (5t + 4)(-t + 5) 5t – 4 = 0 and -t + 5 = 0 5t = 4 and t = 5 t = s and t = 5s

Example #4 cont’d

Can’t have negative time

Therefore, it will take 5s to fall 20m below its

initial height.