Finding the Difference Between Two Numbers

Let the two numbers be denoted by $x$ and $y$.

We are given the following information:

• The sum of the two numbers is 20: $x + y = 20$
• The product of the two numbers is 96: $x \times y = 96$

We need to find the difference between the two numbers, which is $|x - y|$.

Calculating the Difference Using Algebraic Identity

We can use the algebraic identity: $(x - y)^2 = (x + y)^2 - 4xy$.

Substitute the given values into the identity:

$(x - y)^2 = (20)^2 - 4(96)$

$(x - y)^2 = 400 - 384$

$(x - y)^2 = 16$

To find the difference, take the square root of both sides:

$|x - y| = \sqrt{16}$

$|x - y| = 4$

Alternative Method: Finding the Numbers

From the sum equation, $x + y = 20$, we can express $y$ as $y = 20 - x$.

Substitute this into the product equation:

$x(20 - x) = 96$

$20x - x^2 = 96$

Rearrange into a quadratic equation: $x^2 - 20x + 96 = 0$.

Factor the quadratic equation: $(x - 12)(x - 8) = 0$.

This gives two possible values for $x$: $x = 12$ or $x = 8$.

• If $x = 12$, then $y = 20 - 12 = 8$.
• If $x = 8$, then $y = 20 - 8 = 12$.

The two numbers are 12 and 8.

The difference between the two numbers is $|12 - 8| = 4$.

Conclusion

The difference between the two numbers is 4.