Relationship 1 has an initial value of 5 and a rate of
change of 3/2. Relationship 2 has a rate of change of
5/6. Explain how to find what the initial value of
Relationship 2 needs to be for it to have an output of 50
with the exact same input as Relationship 1.

Sample Response: The equation of Relationship 1 is y = 3/2x + 5. Substitute 50 for y and solve to find that x = 30. Multiply 30 by 5/6, the rate of change of Relationship 2, to get 25. This means that the initial value would need to be 25 to get to 50 when the input is 30.

let's call the solutions x₁ = (5 - 2√7)/3 and x₂ = (5 + 2√7)/3

then the equation is (x-x₁)(x-x₂) = 0

if you fill that in you get a lot of numbers which you can recognize as ax²+bx+c=0.

step-by-step explanation:

you can see that a=1 (there is just an x²)

b = (-5 -2√7 + 2√7 - 5)/3 = -10/3

c = -2√7)/3 ) * (-5+2√7)/3) = -1/3

so your answer is 3x^2 – 10x – 1 = 0