Question

The difference between the squares of two numbers is 80. If the sum of the two numbers is 16, what is their positive difference?

Answer 1

Positive Difference between the two numbers is `color(red)5`

Explanation:

Let us assume that the two given numbers are `a and b`

It is given that

`color(red)(a+b=16)` ... Equation.1

Also,

`color(red)(a^2-b^2=80)` ... Equation.2

Consider Equation.1

`a+b=16` Equation.3

`rArr a = 16 - b`

Substitute this value of `a` in Equation.2

`(16-b)^2-b^2=80`

`rArr (256 - 32b + b^2)-b^2 = 80`

`rArr 256 - 32b cancel(+ b^2) cancel (-b^2) = 80`

`rArr 256 - 32b = 80`

`rArr -32b = 80 - 256`

`rArr -32b = - 176`

`rArr 32b = 176`

`rArr b = 176/32`

Hence,

`color(blue)(b=11/2)`

Substitute the value of `color(blue)(b=11/2)` in Equation.3

`a+b=16` Equation.3

`rArr a + 11/2 = 16`

`rArr a = 16 - 11/2`

`rArr a = (32 - 11)/2`

`rArr a = (21)/2`

Hence,

`color(blue)(a = (21)/2)`

Now, we have the values for `a and b`

`color(blue)(a = (21)/2)`

`color(blue)(b=11/2)`

We need to find the positive difference between a and b

We will use absolute value function to find the positive difference.

`|a -b| = |21/2 - 11/2| = |(21-11)/2| = |10/2| = 5`

`:.`Positive Difference between the two numbers is `color(red)5`

Hope this helps.

Answer 2

`5`.

Explanation:

If `x and y` are the reqd. nos., then, by what is given, we have,

`|x^2-y^2|=80, and (x+y)=16..............(ast)`.

But, `|x^2-y^2|=|(x+y)(x-y)|=|x=y|*|x-y|, i.e.,`

`80=16*|x-y|............[because, (ast)]`.

`:. |x-y|=80/16=5`.