Question review: the square root equation with a root that is not there

This is the first of a regular series: one question, worked more than one way, timed. The point is not the answer. It is seeing which method is faster, and how quickly you can tell.

The question

What is the solution to the equation √(2x + 3) = x ?

Four choices, or a box to type in. It appears both ways. It is an Advanced Math question, it shows up in some form on most practice tests, and it has a specific trap that the hand method walks straight into.

By hand

Square both sides to clear the root:

2x + 3 = x²

Rearrange: x² − 2x − 3 = 0. Factor: (x − 3)(x + 1) = 0. So x = 3 or x = −1.

And here is the trap. Squaring both sides can create solutions that were not there. Put x = −1 back into the original: the left side is √(−2 + 3) = √1 = 1, and the right side is −1. They are not equal. x = −1 is extraneous. The only solution is x = 3.

A student who forgets to check picks −1, or picks “both”, and loses the point on a question they solved correctly. In my logs this is a wrong-and-sure answer, fast, confident and wrong, which is the most expensive kind, because the student walks away believing the method works.

Time, done carefully with the check: about seventy seconds.

With Desmos

Left side becomes y=, right side becomes y=:

y=sqrt(2x+3)
y=x

Two curves. One intersection. Click it: x = 3.

There is no second intersection to be tempted by, because the graph of the original equation does not have one. The extraneous root was created by squaring, and the calculator never squared anything. Graphing is immune to the extraneous-solution trap, and this is the best single argument for knowing the intersection method: it does not just save time, it removes an entire category of mistake.

Time: about fifteen seconds, if entering a square root is already automatic.

Which is faster

Desmos, by nearly a minute, and it is also safer. This is a question where the calculator wins on both counts.

But notice why it wins. The hand method is not hard. It is a quadratic and a check. It loses because it has a step that is easy to skip under time pressure, and the skipped step changes the answer. The calculator wins by not having that step at all.

When the hand method still matters

If the choices are written as radicals or fractions rather than decimals, the graph gives you a number you then have to recognize, and that can cost back some of the minute. And if the question is the variation that asks how many real solutions the equation has, you need to know that squaring can invent one, or the graph’s single point will surprise you.

So the by-hand version is worth learning for what it teaches: squaring is not reversible, and every solution to a radical equation has to be checked in the original. The calculator is worth learning because on test day you do not want to rely on remembering that.

What it is actually testing

Not algebra. The College Board writes this question to find out whether you know that a step can add a solution, and whether you check. Students who know the intersection method never meet that trap. Students who do not meet it every time, and about half of them fall in.

If your practice-test log shows this question, or its cousins with rational equations, as right-but-slow or wrong-and-sure, this is the fix, and it takes one session to make it automatic: the intersection method, and practicing the keystrokes in the real calculator.