Question

A positive real number is 4 less than another. If the sum of the squares of the two numbers is 72, then find the numbers.

Answer 1

`2+4√(2)\text{ and }4√(2)-2`

Explanation:

Let the large number be
`x`. We can represent the smaller number with
`x-4`. Since their squares add up to 72, we have the following equation:

`x^2+(x-4)^2=72`

Expand
`(x-4)^2` using the property
`(a-b)^2=a^2-2ab+b^2`:

`x^2+x^2-2(4)(x)+16=72`

Combine like terms:

`2x^2-8x+16=72`

Subtract 72 from both sides:

`2x^2-8x-56=0`

Use the quadratic formula to find solutions for
`x`:

`x=(-b\pm √(b^2-4ac))/(2a)` for
`ax^2+bx+c`

In
`2x^2-8x-56`, assign:

• 
  `a\implies 2`
• 
  `b \implies -8`
• 
  `c\implies -56`

Solving, we get:

`x=(-(-8)\pm √((-8)^2-4(2)(-56)))/(2(2)),\\x=(8\pm 16√(2))/(4),\\\begin{cases}x=(8+16√(2))/(4), x=\boxed{2+4√(2)} \\x=(8-16√(2))/(4), x=\boxed{2-4√(2)}\end{cases}`

Since the question stipulates that
`x` is positive, we have
`x=\boxed{2+4√(2)}`. Therefore, the two numbers are
`2+4√(2)` and
`4√(2)-2`.

Verify:

`(2+4√(2))^2+(4√(2)-2)^2=72\:\checkmark`

Answer 2

Our two numbers are:

`2+4√(2) \text{ and } 4√(2)-2`

Or, approximately 7.66 and 3.66.

Explanation:

Let the two numbers be a and b.

One positive real number is four less than another. So, we can write that:

`b=a-4`

The sum of the squares of the two numbers is 72. Therefore:

`a^2+b^2=72`

Substitute:

`a^2+(a-4)^2=72`

Solve for a. Expand:

`a^2+(a^2-8a+16)=72`

Simplify:

`2a^2-8a+16=72`

Divide both sides by two:

`a^2-4a+8=36`

Subtract 36 from both sides:

`a^2-4a-28=0`

The equation isn't factorable. So, we can use the quadratic formula:

`\displaystyle x=(-b\pm√(b^2-4ac))/(2a)`

In this case, a = 1, b = -4, and c = -28. Substitute:

`\displaystyle x=(-(-4)\pm√((-4)^2-4(1)(-28)))/(2(1))`

Evaluate:

`\displaystyle x=(4\pm√(128))/(2)=(4\pm8√(2))/(2)=2\pm4√(2)`

So, our two solutions are:

`\displaystyle x_1=2+4√(2)\approx 7.66\text{ or } x_2=2-4√(2)\approx-3.66`

Since the two numbers are positive, we can ignore the second solution.

So, our first number is:

`a=2+4√(2)`

And since the second number is four less, our second number is:

`b=(2+4√(2))-4=4√(2)-2\approx 3.66`