Math Is Fun Forum

jadewest

Member

Registered: 2021-02-20

Posts: 44

Solving Quadratic Inequalities

Hi,

Is this the correct way to solve this exercise:

Nathan is a product manager for a home security company.  Through testing and focus groups, he has modeled the equation P = -(x - 32)^2 + 1000 to represent the profit (in thousands) of the price x for a new home security camera.  Find the prices in dollars at which Nathan can make a profit of at least $471, 000.  Show your work as it is done in the lesson content and include units of measurement..  HINT:  Use 471 as part of the inequality you set up, not 471,000.

- (x - 32)^2 + 1000 = 471
1000 - 471 = (x - 32)^2
529 = (x - 32) (x -32)
529 = x (x - 32) - 32 (x - 32)
529 = x^2 - 32x - 32x + 1024
529 = x^2 - 64x + 1024
529 - (x^2 - 64x + 1024) = 0
529 - x^2 + 64x - 1024 = 0
-x^2 + 64x - 495 = 0
- (x^2 - 64x + 495) = 0
x^2 - 64x + 495 = 0

Using the quadratic formula I get the answers:
x = 9 or x = 55

First interval, x < 9, select x = 8
-8^2 + 64(8) - 495 ≥ 471
81 ≥ 471   FALSE

Second interval, 9 < x < 55, select x = 30
2325 ≥ 471   TRUE. x values in that interval are part of the solution

Third interval, x > 55, select x = 56
6225 ≥ 471   TRUE. x values in that interval are part of the solution

ANSWER: 9< x < 55 or x > 55

Is this correct?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Solving Quadratic Inequalities

hi Jade

529 = (x - 32) (x -32)
........

Using the quadratic formula I get the answers:
x = 9 or x = 55

LATER EDIT these are correct.  Apologies.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

jadewest

Member

Registered: 2021-02-20

Posts: 44

Re: Solving Quadratic Inequalities

Hello,

In this exercise I am told to try the three intervals again. What am I doing wrong?

Jade

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Solving Quadratic Inequalities

Hi Jade,

I've had another look at this problem and realised my earlier answer was wrong. Your values, 9 and 55 are correct. Many apologies for that.

So you need to evaluate -(x-32)^2 + 1000 just to the left of 9; just to the right, and again for left and right of 55.  Here's my calculations

Try x = 8.  -(8 - 32)^2 + 1000 = 216 which is less than 471. We need at least 471 so x = 8 is NOT a solution.

Try x = 10       -(10-32)^2 + 1000 = 516 which is more than 471 so this IS a solution.

Try x = 50       -(50 -32)^2 + 1000 = 676 which is more so this IS a solution.

Try x = 60       -(60-32)^2 + 1000 =  216 which is less so this is NOT a solution.

So I'm finding that solutions lie between 9 and 55.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob