Chapter 1.7, Problem 2E

Solution

To describe: The meaning of completing square for an expression of the form $x^2+bx$ .

Given information:

The expression, $x^2+bx$ .

Concept used:

Perfect square formula: $a^2+2ab+b^2={\left(a+b\right)}^2$ ; $a^2-2ab+b^2={\left(a-b\right)}^2$ .

Calculation:

Recall that $p^2+2pq+q^2$ trinomial can be written as a square of binomial as ${\left(p+q\right)}^2$ .

Compare the expression $x^2+bx$ with $p^2+2pq$ . This gives $p=x$ and $2q=b$ or $q=\frac{b}{2}$ .

That means, the expression $x^2+bx$ can be written as a square of binomial by adding the term ${\left(\frac{b}{2}\right)}^2$ .

Conclusion:

Completing the square for an expression of the form $x^2+bx$ is to add the square of the number $\frac{b}{2}$ to the expression, so that the expression can be written as a square of binomial of the form ${\left(x+\frac{b}{2}\right)}^2$ .