A root isn't always a solution.

#5 · A root isn't always a solution

The question. What is the sum of the solutions to (3x² + 2)/(x − 2) = 7x/(x − 2)?

1. 1/3
2. 2/3
3. 2
4. 7/3

The algebra. Both sides have the same denominator, so you only need the numerators: 3x² + 2 = 7x, which is 3x² − 7x + 2 = 0. There are three ways to finish:

• Factor it: (3x − 1)(x − 2) = 0, so x = 1/3 or x = 2.
• Use the quadratic formula: x = (7 ± 5)/6, which gives the same two roots.
• Use the shortcut: the solutions of ax² + bx + c = 0 add up to −b/a, which here is 7/3.

All three give a sum of 7/3.

Or graph it from the start

Type the whole equation into the graphing calculator built into Bluebook. It draws a vertical line at each real solution. You'll see one line, just right of zero, and nothing at x = 2: the calculator never shows the root that doesn't work.

When the solution isn't a whole number

Here the line sits between gridlines, so you can't read it exactly. Graph the two sides separately, y = (3x² + 2)/(x − 2) and y = 7x/(x − 2), then click where they cross. The calculator shows the point (0.333333, −1.4), so x is 1/3.

One limit: the calculator rounds that point, so it can't turn 0.333333 back into 1/3 for you. Typing 0.333333 and switching it to a fraction gives 333333/1000000, so match the decimal to the answer choices yourself.

Where else this happens

Watch for it whenever x is in a denominator, like this question, or under a square root. In both cases the usual algebra can hand you a root that doesn't work in the original equation. So put each root back into the original equation, or graph it.

The takeaway: a root isn't always a solution. Check your roots, or graph the equation.