Question

Find two positive numbers whose difference is 30 and whose product is 2584.

Answer 1

9514 1404 393

Answer:

38 and 68

Explanation:

Let x represent the average of the two numbers. Then their product is ...

(x -15)(x +15) = 2584

x^2 = 2584 +225 = 2809 . . . . . . use relation (x -a)(x +a) = x^2 -a^2; add a^2

x = √2809 = 53

Then the two numbers are ...

53 +15 = 68

and

68 -30 = 38

The two numbers are 38 and 68.

_____

Additional comment

Solving the problem in this way is equivalent to writing the quadratic equation for the smallest (or largest) number and solving that by completing the square. The "solution" x = -53 is not relevant in this problem.

Answer 2

`38` and
`68`.

Explanation:

Let
`x` be the smaller one of the two number.

`x` must be a positive integer. The other number would be
`(x + 30)`.

The question states that the product of the two numbers is
`2584`. In other words:

`x\, (x + 30) = 2584`.

Rearrange this equation and solve for
`x`:

`x^(2) + 30\, x - 2584 = 0`.

The first root of this quadratic equation would be:

`\begin{aligned}x_(1) &= \frac{(-30) + \sqrt{30^(2) - 4 * (-2584)}}{2} \\ &= ((-30) + √(900 + 10336))/(2) \\ &= ((-30) + √(11236))/(2) \\ &= ((-30))/(2) + \sqrt{(11236)/(2^(2))} \\ &= (-15) + √(2809) \\ &= (-15) + 53 \\ &= 38 \end{aligned}`.

Similarly, the second root of this quadratic equation would be:

`\begin{aligned}x_(1) &= \frac{(-30) - \sqrt{30^(2) - 4 * (-2584)}}{2} \\ &= (-15) - 53 \\ &= -68\end{aligned}`.

Since the question requires that both numbers should be positive,
`x > 0`. Therefore, only
`x = 38` is valid.

Hence, the two numbers would be
`38` and
`(38 + 30)`, which is
`68`.