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.

Answer

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Answer

Then $a = \frac{1}{2}$ , so the roots are $- 2, \frac{3}{2}$ .

Steps

Step 1 ：Let the quadratic be $x^{2} + a x + b$ . Then the roots are $a + 1$ and $b + 1$ . By Vieta's formulas,

Step 2 ： $\begin{aligned} (a + 1) + (b + 1) & = - a, \\ (a + 1) (b + 1) & = b . \end{aligned}$

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 $- 2, \frac{3}{2}$ .