square trinomial

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.


x squared +1/2x



What is the constant that should be added to the binomial so that it becomes a perfect square? trinomial?

Answer 1

 

x² + (1/2)x

 

Use the complete the square process

 

    When in the form

       Ax² + Bx

 

           A must be 1.

           take 1/2 of be, square it, and add the result

 

    A = 1, B = 1/2

 

 

     [(1/2)B]² = [(1/2)(1/2)]² = (1/4)² = 1/16

 

You must add 1/16 to make it a perfect square trinomial

 

    x² + (1/2)x + 1/16

=  (x + 1/4)(x + 1/4)

 

 

Answer 2

Any binomial, i.e., an algebraic expression consisting of two terms, after squaring becomes trinomial, an algebraic expression consisting of three terms. Example (a+b)²=a²+2ab+b² 

The given expression = x²+1/2x or x²+(1/2)x which can be written as x²+(x/2). Since the question finally asks for a constant term to be the third term for the given binomil to become a trinomial which will be perfect square, it is evident that the root binomial of the trinomial consists of one variable term which is given as x and one constant term. Hence the given expression can only be x²+(x/2)

 Lets, assume that P, which is a constant, if added to the given expression, then it becomes a the trinomial, x²+x/2+P, which is a perfect square and is = (x+p)²= x²+2px+p² , where p is too a constant

 

In short,  x²+x/2+P=x²+2px+p² 

 

Comparing term by term of L.H.S and R.H.S, x/2= 2px ... (1) and P=p² .... (2)

dividing both sides of (1) by x, 2p=1/2

=>p=1/4... dividing both sides by 2

Substituting, p=1/4 in (2), P=1/16

 

Thus the desired trinomial is  x²+x/2+1/16 and its complete factorization is = (x+1/4)²