Question

Will Give Branliest. To decorate for a party, Rob and Loretta were each given a bag of balloons. There were 40 balloons between both bags. Rob used 8 balloons from his bag, and Loretta used 8 balloons from hers. The product of the number of balloons left in each bag is no more than 44.

Let x represent the number of balloons that Rob had in his bag before he used any. Which choice represents the number of balloons left in their bags?
A. x^2-40x=300
B. x^2-40x+300>=0
C. x^2-40x-300<0
D. x^2-40x-300=0
If Rob originally had more balloons than Loretta, what is the least number of balloons that Rob could have had before using any?
A. 10
B 20
C.30

Answer 1

Last one absolutely(We need quadratic equation not inequality here)

Let's check again for confirmation

$\\ \rm\rightarrowtail x^2-40x+300=0$

$\\ \rm\rightarrowtail x^2-30x-10x+300=0$

$\\ \rm\rightarrowtail (x-30)(x-10)=0$

$\\ \rm\rightarrowtail x=30,10$

X has values 30 and 10

Rob has highest means atleast 30

Answer 2

Answer:

B. x² - 40x + 300 ≥ 0

C. 30

Step-by-step explanation:

Let x = number of balloons Rob started with

Let y = number of balloons Loretta started with

Given:

• There were 40 balloons between both bags

⇒ x + y = 40

Given:

• Rob used 8 balloons from his bag, and Loretta used 8 balloons from hers. The product of the number of balloons left in each bag is no more than 44.

⇒ (x - 8)(y - 8) ≤ 44

Rewrite  x + y = 40  to make y the subject:

⇒ y = 40 - x

Substitute into (x - 8)(y - 8) ≤ 44 and simplify:

⇒ (x - 8)(40 - x - 8) ≤ 44

⇒ (x - 8)(32 - x) ≤ 44

⇒ 32x - x² -256 + 8x ≤ 44

⇒ - x² + 40x - 256 ≤ 44

⇒ - x² + 40x - 300 ≤ 0

Dividing both sides by -1 (remembering to change the direction of the inequality sign):

⇒ x² - 40x + 300 ≥ 0

Factor  x² - 40x + 300 ≥ 0

⇒ x² - 10x - 30x + 300 ≥ 0

⇒ (x² - 10x) + (-30x + 300) ≥ 0

⇒ x(x - 10) -30(x - 10) ≥ 0

⇒ (x - 10)(x -30) ≥ 0

Therefore,  x ≤ 10 or x ≥ 30

As x + y = 40:

If x ≤ 10 then y ≥ 30

If x ≥ 30 then y ≤ 10

If Rob (x) had MORE balloons than Loretta (y) then only
If x ≥ 30 then y ≤ 10 is valid

Therefore, the least number of balloons Rob could have had before using any is 30.