Question

How do you find 2 consecutive even integers whose product is 224?

Answer 1

Two consecutive even integers are `{14,16}` or `{-16,-14}`

Explanation:

Two consecutive even integers could be `n` and `n+2` and as their product is `224`, we have

`n(n+2)=224` or `n^2+2n=224` or

`n^2+2n-224=0`

hence `n=(-2+-sqrt(2^2-4*1*(-224)))/(2*1)` or

`n=(-2+-sqrt(4+896))/(2*1)=(-2+-sqrt900)/2=(-2+-30)/2`

Hence `n=(-2+30)/2=14` or `n=(-2-30)/2=-16`

Hence two consecutive even integers are `{14,16}` or `{-16,-14}`

Answer 2

`(14,16)and(-14,-16)`

Explanation:

Let the first integer may be `n`

Remember that even numbers differ in `2`

So,the second number will be `n+2`

`color(purple)( :.n(n+2)=224`

Use distributive property `color(brown)(a(b+c)=ab+ac`

`rarrn^2+2n=224`

`rarrn^2+2n-224=0`

This is a Quadratic equation (in form `ax^2+bx+c=0`)

Use Quadratic formula

`color(brown)(x=(-b+-sqrt(b^2-4ac))/(2a)`

Where

`color(red)(a=1,b=2,c=-224`

`rarrx=(-2+-sqrt(2^2-4(1)(-224)))/(2(1))`

`rarrx=(-2+-sqrt(4-(-896)))/(2)`

`rarrx=(-2+-sqrt(4+896))/(2)`

`rarrx=(-2+-sqrt(900))/(2)`

`rarrx=(-2+-30)/(2)`

Now we have two solutions

`color(indigo)((-2+30)/(2)=28/2=14`

`color(violet)((-2-30)/(2)=-32/2=-16`

`n` is expressed here as `x`

`:.n=(14and -16 )`

So, the integers are `color(green)((14,16)and(-14,-16)`