Difference of Squares Example Problems

Try the the following 5 sample questions, hand-picked by the Target Test Prep GRE experts. When solving, use the difference of squares formula to answer each question strategically.

Example 1

medium

Which of the following is equivalent to a100  b100a50  b50 for all values of a and b for which the expression is defined?

a2 + b2

a2 – b2

a50 + b50

a50 – b50

(ab)2

Confirm your answer

Solution:

Notice that a100b100 is in the form of a difference of squares because a100 is the square of a50 and b100 is the square of b50.

Thus, a100b100=(a50)2(b50)2=(a50b50)(a50+b50).

a100b100a50b50=(a50)2(b50)2a50b50=(a50b50)(a50+b50)a50b50=a50+b50

Correct answer:C

Example 2

easy

What is the value of (555)2(55)2500?

6,100

5,555

5,500

655

610

Confirm your answer

Solution:

It would take a significant amount of time to square the values in the numerator and then subtract them. An easier approach is to notice that the numerator is in the form of a difference of squares:

x2y2=(x+y)(xy). Here, x = 555, and y = 55.

(555)2(55)2500(555+55)(55555)500(610)(500)500610

Correct answer:E

Example 3

hard

The radius, R, of a larger circle is 555 centimeters. The radius, r, of a smaller circle is 55 centimeters. Which of the following properly expresses, in square meters, the difference between the areas of the two circles? (Note: 1 meter = 100 centimeters)

30.5π

305π

3050π

30,500π

305,000π

Confirm your answer

Solution:

The difference of the areas can be expressed as πR2πr2. We can use the difference of squares to help us easily subtract these quantities. 

πR2πr2π(R2r2)π(Rr)(R+r)π(55555)(555+55)π(500)(610)305,000π

305,000π is the difference between the areas of the two circles in square centimeters. However, we want to know the difference between the areas in square meters. Since there are 100 centimeters in 1 meter, there are 1002 = 10,000 square centimeters in 1 square meter.  Therefore, the difference between the areas of the two circles, in meters, is: 30.5π.

Correct answer:A

Example 4

medium

On Monday an item was marked up x percent. On Tuesday it was marked down x percent. If the original price of the item was d dollars, which of the following correctly expresses the final price of the item after the markup and markdown?

d

d(x10010,000)

d(x100)

100d(x210,00010,000)

d(10,000  x210,000)

Confirm your answer

Solution:

To mark the item’s price up x percent, we must remember that we are increasing the item’s original price of d dollars by x percent. The percent increase function is expressed as (1+x100). Notice that without adding 100% to the x percent, we would be reducing the item’s price by (100 – x) percent. Similarly, when reducing the item’s resulting price by x percent, we must multiply this new starting price by (1x100). The item’s ending price is therefore

d(1+x100)(1x100)    d(100+x100)(100x100)d(1002x210,000)d(10,000x210,000)

As we can see in the calculation above, there is a difference of squares of (100+x)(100x) in the numerators of the fractions. We can therefore avoid the standard method of simplifying a quadratic equation given the property (100+x)(100x)=(1002x2).

Correct answer:E

Example 5

medium

Set S = {x, 14x, y2, 5y, x2, y}

If in set S above, x is an integer greater than one and 0 < y 0.7, which of the following represents the range of the set S?

(x + y)(x − y)     

x​– 3y  

y2 

x – y

x− y

Confirm your answer

Solution:

Our job is to determine which terms represent the largest and smallest values in the set. Since three of the six terms involve x and the other three involve y, we need to find the largest and the smallest terms that involve x and the largest and the smallest terms that involve y.

We know that x is an integer greater than one, so it must be positive. Out of all the values for terms involving x, x2 must be the greatest because the square of an integer greater than one will always be greater than the integer, and 14x must be the smallest since 14 of a positive number is less than the number. For example, if x = 2 (the smallest integer greater than 1), then x2 = 4, and 14x=0.5. Notice that 4 and 0.5 are the smallest values of x2 and 14x, respectively.

Next, we know that y is a positive decimal less than or equal to 0.7. Out of all the values for terms involving y, 5y must be the greatest because 5 times a positive number is greater than the number, and y2 must be the smallest since the square of a positive decimal less than one will always be less than the decimal. For example, if y = 0.7 (the largest decimal it can be), then 5y = 3.5 and y2 = 0.49. Notice that 3.5 and 0.49 are the largest values of 5y and y2, respectively.

As we can see, for the term with the largest value, we should either choose x2 or 5y since they are the greatest of the terms involving x or y, respectively. However, we should choose x2 since the smallest possible value of x2 is 4, which is greater than 3.5, the largest possible value of 5y. Similarly, for the term with the smallest value, we should either choose 14x or y2 since they are the smallest of the terms involving x or y, respectively. However, we should choose y2 since the largest possible value of y2 is 0.49, which is less than 0.5, the smallest possible value of 14x.

Consequently, the range equals x2 – y2, which in this case is represented as the difference of squares in factored form as (x + y)(x – y).

Correct answer:A