Question

Explain why completing the square of the expression \(x^{2}+b x\) is easier to do when \(b\) is an even number.

Short answer

Completing the square is easier when \(b\) is even. This is because halving \(b\) (step in completing the square) results in an integer when \(b\) is even, leading to a simpler process, while it results in a fraction or decimal if \(b\) is odd, making the process more complex.

Step-by-step solution

Understanding the process of completing the square

To complete the square for the given quadratic expression \(x^{2}+b x\), first rewrite the expression in this form: \(x^{2}+b x +\left(\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2\). This is done because \(\left(\frac{b}{2}\right)^2\) is the value that makes the expression a perfect square trinomial, \(x^{2} + b x +\left(\frac{b}{2}\right)^2\), which can be rewritten as \(\left(x+ \frac{b}{2}\right)^2\). The \(-\left(\frac{b}{2}\right)^2\) was added to balance the equation since the same value was subtracted.

Completing the square when b is even

When \(b\) is even, the halving process (\(b/2\)) results in an integer. This makes the square completion process cleaner and simpler. There are fewer fractions, making the calculation easier to understand and perform.

Completing the square when b is odd

When \(b\) is an odd number, halving \(b\) results in a decimal or fraction. This results in a more complex square completion process which includes fractions. Often, these fractions result in more steps or more complex arithmetic.