Astudent brings whole cherry and cheese danishes to his class for his birthday. the number of cherry danishes he brings is at least 3 more than the number of cheese danishes, but no more than twice the number of cheese danishes. find the smallest possible value for the total number of danishes he brings.

at least 8

Step-by-step explanation:

Let the variable $r$ represent the number of cherry danishes and $s$ represent the number of cheese danishes. We have$$2s \ge r \ge 3+\frac{2}{3}s.$$The left side of the inequality must be greater than or equal to the right side, so we have$$2s \ge 3+\frac{2}{3}s\qquad

\Rightarrow\qquad 6s \ge 9+2s \qquad

\Rightarrow\qquad 4s \ge 9 \qquad

\Rightarrow\qquad  s \ge \frac{9}{4}.$$The least possible integer that $s$ can be is 3. The least possible value of $r$ is$$r \ge 3+\frac{2}{3}(3) \qquad \Rightarrow \qquad r \ge 5.$$Therefore, the student brings at least 8 danishes

* credits: AoPS

the polynomial is a perfect square trinomial of the form a^2-2ab+b^2:

option a. true

step-by-step explanation:

9x^2-12x+4

comparing with the form a^2-2ab+b^2

1.) a^2=9x^2

solving for a: square root both sides of the equation:

sqrt(a^2)=sqrt(9x^2)

a=sqrt(9) sqrt(x^2)

a=3x

2.) b^2=4

solving for b: square root both sides of the equation:

sqrt(b^2)=sqrt(4)

b=2

to be a perfet square trinomial 2ab must be equal to 12x. let's check:

2ab = 2 (3x) (2)

2ab = 12x

correct

for this case we must apply distributive property to the following expression:

the distributive property states that:

then, when applying the property we have to multiply the -2 for each term within the parenthesis: