1. Quadratic equations and prime numbers
The general quadratic equation is given by the following equation.
In the above equation, .If A = 0, the above equation no more remains quadratic equation; rather it comes single order equation and its solution is given by.
There are many ways to solve (1) but using quadratic formula is quite popular as we can solve any quadratic equation by using this formula. We will derive this formula here.
The approach in deriving quadratic formula is first to complete the whole square in left hand side after transferring the constant term to right hand side. Take the square root of the whole equation. First, consider the +ve square root and find out the first root of the equation. Second, consider the –ve square root and find out the second root of the equation. There will be two roots of the quadratic equation. These can be distinct and real roots, repeated root or complex conjugate roots depending upon the nature of the square root on right hand side. This is basic approach for deriving the quadratic formula.
The various steps are given below
a) Take constant term C to the RHS and the equation becomes as given below.
b) The coefficient of x2 term is A. Hence, we multiply the whole equation by 4A and get the following equation.
c) The square of coefficient of x is B and its square is B2. We add it on both sides of the above equation and get the following.
d) The left hand side of the above equation is . Hence, we can write the above equation as given below.
We take square root on both sides of the equation and get the following.
e) We consider +ve sign on RHS and get the following equation.
Solving this, we get
This is one root of the quadratic equation.
f) We consider -ve sign on RHS and get the following equation.
Solving this, we get
This is second root of the quadratic equation.
Thus, we can find out roots of the quadratic equation if we know the values of A, B and C.
2. PRIME NUMBERS
A number which is divisible by only 1 and itself is called prime number. For example, 3 is prime number as it is divisible by 1 and 3 only. If we substitute some integer value of x in expression , we get prime number. We make the following table.
Thus, we get prime number by substituting various values of x in the expression .
Let us suppose that on substituting some value of x, we get composite number 8. Let us find out the corresponding value of x. Hence, we have
Thus, the value of x is complex. Hence, we don’t get real value of x which will give us complex number on substituting in the expression . We can try any other composite number also and ensure that we cannot find real value of x which will give us composite number on substituting in the expression .