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Plugging In: Square Root Equations On the SAT Math Section

Many of my students start studying with me after a period with unsatisfactory prep courses or private tutors. And they often bring the bad advice they've collected along the way with them. One such piece of bad advice is "plugging in" answer choices in the math section. Some of my students actually think of this as their go-to method: if you can plug in, you should plug in.

In actuality, plugging in is usually a terrible way of solving math questions on the SAT. It should only be used if you're unprepared for your exam and can't answer the question directly. In that case, at least plugging in will deliver an answer, even if that answer does take a while.

Why you SHOULDN'T plug in (usually)

In the best case scenario, you're spending too much time on a question that could be answered much more easily. In the worst case scenario, you're plugging in numbers in a question which can't be solved by plugging in. Many of my students don't grasp the logical premise of a question enough to know when plugging in can, at least theoretically, solve a question, and when it can't.

Take, for example, the following example:

Question #1

Which of the following solves for the equation (x-3)² = 25 ∕ 11?

A) 28 / 11, 7 / 11

B) 38 / 11, - 38/11

C) √7 / √11, - √7 / √11 D) 38 / 11, - 28 / 11

How long do you think it would take to plug in these options? What about if you hit the jackpot on your fourth try?

There's a way to solve this question within 30 seconds. (Hint: it does not involve FOIL.) When you practice enough, you'll see the proper method immediately, and it should take about a quarter of the time it would by plugging in.

Now, not all questions will involve options as ugly as these. But the average question will have at least some nasty values, and it could easily be a few minutes before you're done. No question on the SAT should take two minutes. If you've allowed yourself a respectable amount of time to prepare for the SAT, and you're serious about getting an excellent score, plugging in is almost never the way to go.

Almost never.

When SHOULD you plug in? There are two general cases (there are exceptions) when plugging in is acceptable:

1. Systems of equations

2. Square root equations

Now, there's a fundamental difference between these two things. For case (1), you can choose to plug in if you so choose. These equations tend to be fairly simple, and so are the numbers given as options.

My personal recommendation would be to solve. The greater your algebraic skill, the better you will be at solving the entire math SAT section. When you regularly plug in during practice sessions instead of developing those skills, you weaken your abilities.

However, neither of these methods - plugging in or solving directly - is inherently faster than the other. It's a matter of personal style and preference. Some will do one method quicker, some the other.

For case (2), however, the situation is drastically different. There, it is almost always faster to plug in.

Let's look at an example.

Example #1

What is the solution set of the following equation:

√(2x-5) - 3 = 1 - x

Question #1

A) {7}

B) {3}

C) {7, 3, -1}

D) {7, 3}

The inherent problem in a square-root equation

There's a direct way to solve this question. Firstly, numbers have to be shifted so that the square root √(2x-5) is on one side and everyone else is on the other. Next, both sides of the equation have to be squared. At this point, you'll be dealing with a quadratic equation; solve it and get two solutions. Next, check the two solutions. One of them is most likely incorrect.

Why would a solution be incorrect, if the equation was solved correctly?

The original equation was not a quadratic equation at all, but rather a square root problem. This comes with the inherent implication that the value inside the square root is positive. When you square both sides, that "information" is lost, resulting in a solution which doesn't solve the original equation.

Don't worry if the above explanation seems unclear. The gist is that, if you solve these questions directly and correctly, you'll likely end up with a wrong answer anyway. A unique characteristic of square root problems on the SAT, is that solving them is more difficult and error-prone than plugging in.

Now let's look what happens when you plug in.

Note that these are easy equations to plug into. And the values in this question type also tend to be simple - no huge numbers, radicals, or fractions. In addition, because we're working with sets, there aren't 7 different numbers to plug in as first appears, but rather only three numbers that really have to be checked: 7, 3, and -1.

-1 leads to the radical of a negative number, and so causes the question to be undefined. 7 leads to two different values on either sides. 3 works. Option B) is the answer.

Let's look at another example:

Example #2

Given: (7+x) / √(x²-4)

Question #2

Which is the following is the domain of the above equation?

A) x ≥ 2, or x ≤-2

B) -2 ≤ x ≤ 2

C) x > 2, or x <-2 D) -2 < x < 2

'Domain' is the set of x where the function is valid. The laws of domains can be technical and confusing, and the inequality here makes it worse. Let's try plugging in and see what happens.

The most obvious value to plug in is either 2 or -2. Since positive numbers are usually simpler to work with, we'll plug 2 into the given equation. No good: we end up dividing by zero. 2 can't be in the answer range, which eliminates options A) and B).

We're left with options C) and D). 0 is in the range of option D) but not that of C), which will help us filter out the answer. We'll plug in 0 to the equation and end up, in the denominator, with the radical of a negative number. Invalid.

The answer is option C).

Let's look at one final example:

Example #3

Given the equation:

√(2x+k) = x-3

Question #3

If k = 9, which of the following is the solution set of the equation?

A) {8}

B) {0}

C) {0, 8}

D) no solution

Plug in, and you'll get that only 8 works. Option A) is the answer.

Note that in a function or in an equation, the square root is always positive. When you plug in 0 to this equation, you'll get √9 on one side, and -3 on the other. But in an equation, the square of a number is always its positive value. √9 equals 3, not -3. Therefore, 0 would not solve this equation, and options B) and C) are out. We'll discuss this topic further in a future article.

Concluding notes

For the vast majority of questions, avoid plugging in. If you wish to do so for systems of equations, you can, though it's preferable to practice solving them. For questions with radicals, the best option is almost always to plug in. Radical questions like those shown above have about a 50 / 50 probability of appearing on your own SAT.

For more tips and methods, or to get help personally tailored to your needs, consider working with me. I've helped people from all over the world get into their dream school. Studying privately with a skilled teacher is the best way to increase your test score, and in coaching you, I adopt my methods specifically to your personality, schedule, and learning style.

Happy learning,