Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}+20 x$$

Short answer

The term that should be added to the expression to create a perfect square trinomial is 100.

Step-by-step solution

Identify the structure of a perfect square trinomial

A perfect square trinomial is of the form \(a^{2}+2ab+b^{2}\). It can be made by squaring a binomial (a+b)^{2} or (a-b)^{2}.

Write the given expression as a uncompleted square trinomial

We have \(x^{2}+20x\). Our aim is to find the term which, when added to this expression, will complete the square. This will be equivalent to finding the 'b^{2}' term in our perfect square trinomial. The 2ab term is equivalent to 20x. Therefore, by substituting x for a and comparing terms, we see that 2ab = 20x which simplifies to 2b = 20.

Calculate the value of b

Solving our equation for b gives us b = 20/2 = 10.

Calculate the perfect square term

Now that we have the value of 'b', we can find the perfect square term 'b^{2}', which is going to be 10^{2} = 100 which will be added to the expression to make it a perfect square trinomial.

Validate the perfect square trinomial

Insert the term into our original binomial to confirm that we have created a perfect square trinomial: \(x^{2} + 20x + 100\) is indeed \((x+10)^{2}\), a perfect square trinomial.