Question:

find the smallest of 3 consecutive positive integers such that when 5 times the largest is subtracted from the square of the middle one the result exceeds three times the smallest by 7.

Ans:



Let the numbers be n and (n+1) and (n+2)



(You can also assume n-1, n and n+1 - the approach is still the same)



Now let's look at each part of the question and translate it into algebraic form.



Largest number = n+2

5 times the largest number =  = 



Middle number = n+1

Square of the middle number = 



Smallest number = n

3 times the smallest number = 



"5 times largest number" subtracted from "square of middle number" will give you



 = 



This value "exceeds '3 times the smallest' by 7" means this is equal to 

"3 times the smallest + 7" i.e. it is 



Equating the above, 



i.e. 

This is a standard quadratic equation, which can be solved by factorization or

by the quadratic formula, as shown below.



The solutions are n = 8, or n = -2



Since it is given that the numbers are positive, the set of 3 numbers are

8,9 and 10.



Check: Square of middle number - 5 times largest number = 81 - 50 = 31

3 times smallest number = 3 * 8 = 24.

31 exceeds 24 by 7 - correct!



:)



 
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation  (in our case ) has the following solutons:

  

  

  

  For these solutions to exist, the discriminant  should not be a negative number.

  

  First, we need to compute the discriminant : .

  

  Discriminant d=100 is greater than zero. That means that there are two solutions: .

  

    

    

    

    Quadratic expression  can be factored:

  

  Again, the answer is: 8, -2.
Here's your graph:







 

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
 find the smallest of 3 consecutive positive integers such that when 5 times the 

largest is subtracted from the square of the middle one the result exceeds three times the smallest by 7.



i cant seem to figure out what "exceeds three times the smallest by 7" part. i get the other part in front. i just need help translating what that part means into algebraic form and i then i can solve the equation.





Let the first of the 3 consecutive positive integers be F

Then the second, or middle integer is F + 1, and the largest is F + 2. I hope you're with me so far!!





 --------- Translating "when 5 times the largest is subtracted from the square of the middle one the result exceeds three times the smallest by 7."





Note that "when 5 times the largest is subtracted from the square of the middle one,"5 times the largest" or 5(F + 2) is subtracted from the square of the middle one () by placing 5(F + 2) after the . 





Additionally, note that  is LARGER than 3 times the first integer (3F) by 7, so  

RELATED QUESTIONS

Find the smallest of 3 consecutive integeres such that when 5 times the largest is... (answered by jorel1380)

Find three consecutive positive integers such that the cube of the smallest minus the... (answered by CubeyThePenguin)

Find four consecutive odd integers such that the sum of the two largest integers... (answered by ewatrrr)

Find four consecutive odd integers such that the sum of the two largest integers... (answered by CubeyThePenguin)

Find four consecutive odd integers such that the sum of two largest integers subtracted... (answered by josmiceli)

find four consecutive odd integers such that the sum of the two largest integers... (answered by CubeyThePenguin)

 Find three consecutive even integers such that if the largest integer is subtracted from  (answered by ankor@dixie-net.com)

find three consecutive integers such that the square of the smallest increased by four... (answered by Theo)

find three consecutive positive integers, such that the square of the smallest exceeds... (answered by mananth)