Suppose the roots of the polynomial $x^2 - mx + n$ are positive prime integers (not necessarily distinct). Given that $m < 20,$ how many possible values of $n$ are there?

18 values for n are possible.

Step-by-step explanation:

Given the quadratic polynomial:

such that:

Roots are positive prime integers and

To find:

How many possible values of are there ?

Solution:

First of all, let us have a look at the sum and product of a quadratic equation.

If the quadratic equation is:

and the roots are: and

Then sum of roots,

Product of roots,

Comparing the given equation with standard equation, we get:

A = 1, B = -m and C = n

Sum of roots,  

Product of roots,

We are given that  

and are positive prime integers such that their sum is less than 20.

Let us have a look at some of the positive prime integers:

2, 3, 5, 7, 11, 13, 17, 23, 29, .....

Now, we have to choose two such prime integers from above list such that their sum is less than 20 and the roots can be repetitive as well.

So, possible combinations and possible value of are:

So,as shown above 18 values for n are possible.

30 units^2

step-by-step explanation:

area is l × w

the length is 3

and the width is 10

3 × 10= 30 units squared

step-by-step explanation:

you can add the two numbers and divide the result by 2. or you can find the gap between the two numbers and add half of this to the smaller number.