College Algebra Word Problem

Question

The sum of the squares of three consecutive positive integers is 2030. What is the middle integer?

Answer 1

x, x+1, x+2

x^2+x^2+2x+1+x^2+4x+4=2030

3x^2+6x+5=2030

3x^2+6x+5-2030=0

3x^2+6x-2025=0

x^2+2x-675=0

675=3*3*3*5*5=27*25 and 27-25=2

factor

(x+27)(x-25)=0

x-25=0

x=25

x+1=26

x+2=27

you can get the answer using arithmetic...

2030/3=676.666...

√676.666...=26.012817...

there is your middle number of 26 !!

 

Answer 2

So, Tyler, let's let a = positive integer #1. Then the second is a + 1 and the third is a + 2, right?

 

If I square a, I get a². If I square a + 1, I get a² + 2a + 1. If I square a + 2, I get a² + 4a + 4.

 

Since they all add to 2030, then a² + a² + 2a + 1 + a² + 4a + 4 = 2030

 

Combining my like terms, I get 3a² + 6a + 5 = 2030

 

Subtracting 2030 from both sides, I get 3a² + 6a - 2025 = 0

 

Since 3, 6, and -2025 are all divisible by 3, I can divide everything by 3.

 

a² + 2a - 675 = 0

 

Factoring out, we get (a + 27)(a - 25) = 0

 

Sooooo, a + 27 = 0, and a = -27

 

Aaaaand a - 25 = 0, so a = 25

 

Since we are looking for positive integers, your answer is 26.

 

Hope this helps.