Problem 40 - Algebra and Trigonometry

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. $$ x^{2}+5 x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial is 6.25. The binomial becomes a perfect square trinomial \(x^{2}+5x+6.25\). The trinomial is factored as \((x + 2.5)^{2}\)

Step 1: Identify the a and b values in the binomial

The given binomial has the form \(x^{2}+5 x\). This is similar to the trinomial which is of form \(x^{2}+2ab+b^{2}\). Compare the binomial with the trinomial form, we can see that \(a=x\) and \(2ab=5 x\). So, we can solve for \(b\), \(b=(5 x)/(2*x)=2.5\).

Step 2: Find the missing term for the perfect square trinomial

Now that we have a and b, we can now find the missing term that makes the binomial a perfect square trinomial. The missing term is \(b^{2}\), that is, \((2.5)^{2}=6.25\).

Step 3: Write and factor the perfect square trinomial

The missing term \(6.25\) is added to the original binomial to make the perfect square trinomial, and we have \(x^{2}+5x+6.25\). This trinomial can be factored into \((x + 2.5)^{2}\).