Problem
The equation $x^{2}-8 x+5=0$ can be solved using more than one method. What characteristic of the equation indicates that each method below is or is not a valid technique to use? Since the middle term is an even number, taking half and squaring would give a whole number, which makes [ Select ] [ Select ] method to use. Since the expression is not factorable, the [Select] is [ Select ] method to use. This equation is trinomial so [ Select ] [ Select ] a good method to use. This trinomial does not factor so [ Select ] is not an option.
Solution
Step 1 ：Let the quadratic be \(x^2 + ax + b\). Then the roots are \(a + 1\) and \(b + 1\). By Vieta's formulas,
Step 2 ：\begin{align*} (a + 1) + (b + 1) &= -a, \\ (a + 1)(b + 1) &= b. \end{align*}
Step 3 ：From the first equation, \(a + 1 = -\frac{b}{2}\). Substituting into the second equation, we get
Step 4 ：\[-\frac{b}{2} (b + 1) = b.\]
Step 5 ：Since \(b\) is non-zero, we can divide both sides by \(b\), to get \(-\frac{1}{2} (b + 1) = 1\). This leads to \(b = -3\).
Step 6 ：Then \(a = \frac{1}{2}\), so the roots are \(\boxed{-2,\frac{3}{2}}\).
