The sum of two numbers is 20 and their product is 96. What is the difference between the two numbers?

Correct option is A

Given:

The sum of two numbers is S=20

The product of the two numbers is P=96
We need to find the difference between the two numbers.

Formula Used:

​$$(a-b)^2 = (a+b)^2 - 4ab \\$$​​

Solution:

Let the number be a and b

(a+b) = 20 and ab = 96

​$$(a-b)^2 = (a+b)^2 - 4ab \\a-b= \sqrt{ (a+b)^2 - 4ab}\\ a-b= \sqrt{ 20^2 - 4\times 96}\\ a-b= \sqrt{ 400 - 384}\\ a-b = 4$$​​

Alternate Method:

The numbers can be found using the quadratic equation:

​$$t^2 - (x+y)t + (x \cdot y) = 0$$​​

Substituting the given values:

​$$t^2 - 20t + 96 = 0$$​​

Solve for t using the quadratic formula:

​$$t = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(96)}}{2(1)}$$​​

t = $$\frac{20 \pm \sqrt{400 - 384}}{2} ​​$$​​

t= $$\frac{20 \pm \sqrt{16}}{2}$$​​

t =$$\frac{20 \pm 4}{2}$$​​

t=$$\frac{20 + 4}{2} = \frac{24}{2} = 12$$​​

t= $$\frac{20 - 4}{2} = \frac{16}{2} = 8$$​​

Thus, the two numbers are 12 and 8.
The difference between them is:

12−8=4

Option (a) is right.

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