Using Substitution to Simplify

By Katharine Rudzitis

On the GMAT, you’ll sometimes encounter tough mathematical expressions. Consider the following Data Sufficiency question:

What is the value of x?

Statement 1: (x² – 3)² - 4(x² – 3) – 12 = 0

Statement 2: x² – 2x – 3 = 0

Let’s begin with statement 1: (x² – 3)² - 4(x² – 3) – 12 = 0

Some people will instinctively want to expand, simplify, factor, and eventually solve this equation. If you were to push ahead and solve this equation without looking for a shortcut, you’d have to expand and simplify the left-hand side, which would result in a quartic equation (an equation featuring a variable raised to the power of 4). That’s beyond what the GMAT expects students to solve, so there must be an easier way.

Before we begin applying a variety of algebraic techniques to the left side of this equation, let’s take a moment to look for an easier way to simplify it.

Look for a New “Variable”

Take a second glance at the equation:

(x² – 3)² - 4(x² – 3) – 12 = 0

There’s a repeated term: notice that (x² – 3) appears twice. What if we replace that expression with a new variable?

If we let u = (x² – 3), then our equation becomes u² - 4u – 12 = 0

This equation is much easier to solve. To factor the left side, we’ll need two numbers that add to -4 and multiply to -12. The only numbers that work are -6 and 2, so we get

(u – 6)(u + 2) = 0.

So, either u – 6 = 0 or u + 2 = 0, which means u = 6 or -2.

We’re not done yet! We have to use this information to find the value of x.

Don’t forget that u = (x² – 3)

If x² – 3 = 6 then x² = 9, which means x = 3 or -3

If x² – 3 = -2 then x² = 1, which means x = 1 or -1

Since there are four possible values of x, statement 1 is not sufficient.  

Now onto statement 2: x² – 2x – 3 = 0

Thankfully this is a straightforward quadratic equation.

Factor the left side to get: (x – 3)(x + 1) = 0

So, x = 3 or x = -1

Since statement 2 yields two possible values of x, it is not sufficient. 

The statements combined

Statement 1 tells us that x = 1, -1, 3 or -3

Statement 2 tells us that x = -1 or 3

So, when we combine the two statements, we see that x can equal either -1 or 3.

Since we still can’t determine the value of x, the combined statements are not sufficient, so the correct answer is E.