Desmos for the SAT · Technique 13 of 23

Check if Two Expressions Are Equivalent

Graph both, and if they land on the exact same curve, they are equal.

SolvesEquivalent expressionsSimplifyingFactoring checks

Watch the lesson

The whole move worked on one real SAT question, narrated.

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A second example: two expressions graphed on top of each other, and the invisible hole that makes the condition necessary.

Some SAT questions hand you a messy expression and ask which of the choices is equivalent to it. The by-hand way is to factor, expand, and cancel until you match a choice, and every one of those moves is a spot to drop a sign or cancel a term you should not have.

There is a faster check: graph the original expression as one curve, graph your candidate answer as a second curve, and see if they land exactly on top of each other. If two expressions are equal for every x, their graphs are the same curve. A wrong option draws its own curve the moment you graph it.

One catch: this only works for single-variable expressions, the kind you can graph as y = (expression). A two-variable expression is a surface, not a 2D curve, so this move does not apply there.

Nova

Here's a move that turns 'which one is equivalent?' into a quick eye test: graph the original, graph your guess, and see if they land on the same curve. When they overlap, you're done. No factoring, no canceling.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps factoring a quadratic and canceling by hand.

The question

(2x² + 7x - 15) / (x + 5)

Which of 2x - 3, 2x + 3, x - 3 and 2x - 5 is equivalent to it?

The algebra way

Factor the numerator: 2x^2 + 7x - 15 has to split as (2x - 3)(x + 5), and finding that pair is the slow part.
Cancel the shared (x + 5), leaving 2x - 3.
Pick the wrong factor pair, or cancel a term instead of a whole factor, and you land on a trap choice.

The Desmos way

Type y = (2x^2 + 7x - 15) / (x + 5) and y = 2x - 3 as two lines.
One line appears where you expected two, so the two expressions match at every x.

The payoff: Graphing both skips the factor-pair hunt entirely, which is exactly where the trap answers are waiting.

Step by step

The whole move, all 4 steps

Type the original expression as its own line, y = (original expression).
Type the answer choice you are testing as a second line, y = (that choice).
Look at whether the two curves land exactly on top of each other, one hiding the other.
If the choice separates onto a curve of its own, rule it out and test the next one.

The question

The expression $\frac{2x^{2}+7x-15}{x+5}$ is given, where $x \neq -5$ .

Which of the following is equivalent to the expression?

A $2x - 3$
B $2x + 3$
C $x - 3$
D $2x - 5$

Step 1 of 4

Type the original expression as its own line: y = (2x² + 7x - 15)/(x + 5). Type it straight through. Once you press /, everything after it goes into the denominator, and nothing comes after the denominator here, so you never need the right arrow.

In the calculator: y=(2x^2+7x-15)/(x+5)

Copy it in exactly as the problem gives it, fraction bar and all. No factoring first, that is the whole point.

Now try it on a real SAT question

Head or scratch paper. Distribute the minus sign and collect: faster on paper than in the calculator. Graph both afterwards if you want the check, because the whole risk here is the sign on that $-2x$ .

The expression $(9x^{3} + 5x) - (4x^{3} - 2x)$ is given.

Which expression is equivalent?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

A two-variable expression graphs as a surface, not a 2D curve, so this graph-both-and-compare check does not apply to it. Rearrange it by hand.
A removable hole can make two curves look identical even when they differ at one point. The walkthrough at the top of this page has one: the original is undefined at $x = -5$ while $2x - 3$ is fine there, and the gap is too small to see. That is why SAT choices carry conditions like $x \neq -5$ .

Nova

This is a single-variable move. If the expression has two variables it graphs as a surface, not a 2D curve, so this check does not apply; give the letters numbers and test the choices instead. And a removable hole, one x-value where the original is undefined, is too small to see, so two curves can look identical and still differ at that one point. That is why the question lists the excluded values.

All Desmos techniques

Put this move into real practice

"Check if Two Expressions Are Equivalent" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.