Sum of the squares of two consecutive even integers is 100. Find them

Question

Quadratic equation problem

Answer 1

the sum of two consecutive even integers is 100

10²=100

the consecutive numbers must be less than 10

what are the choices ?

8 and 6

6 and 4

4 and 2

8²=64 and 6²=36

64+36=100

6 and 8 are the integers

if you want negative integers also use -6 and -8 as well

Answer 2

Hi Alex,

 

We have 2 consecutive EVEN integers.

 

Let's define them as 

X and X+2

 

From the first sentence we know

 

X² + (X+2)² = 100

 

Expanding this we have

 

X² + (X²+4X+4) = 100

2X² + 4X + 4 = 100

2X² + 4X + 4 -100 = 0

2X² + 4X - 96 = 0

 

Divide by 2

x² + 2x - 48 = 0

 

And factor

(x+8)(x-6) = 0

 

Set each factor equal to 0 and solve for X.

 

X + 8 = 0    X = -8

X - 6 = 0     X = 6

 

We need to check these two solutions and see if they meet the initial condition.

 

if X=-8, then we have

 

(-8)² + (-8+2)² = 64+(-6)² = 64+36=100

 

if X=6, then we have

(6)² + (6+2)² = 36+(8)2 = 36+64=100

 

Both solutions work.

 

So the integers are 6 & 8 OR -8 & -6.