This is a difference of squares. The difference of squares formula is _a_2 – _b_2 = (a + b)(a – b).

In this problem, a = 6_x_ and b = 7_y_:

36_x_2 – 49_y_2 = (6_x_ + 7_y_)(6_x_ – 7_y_)

This can be solved by looking at all of the answers and multiplying them and comparing to the answer. However this is time consuming. You can start by noting that the term can be a result of , where one of the terms is negative, so one answer can be eliminated. It is also clear that must be the result of multiplied by , so two additional answers can be eliminated. Looking at the last two answers and multiplying through, the correct answer can be determined.

Because the term has a coefficient, you begin by multiplying the and the terms () together: .

Find the factors of that when added together equal the second coefficient (the term) of the polynomial: .

There are four factors of : , and only two of those factors, , can be manipulated to equal when added together and manipulated to equal when multiplied together:

Before we do anything, we notice that both terms in the expression have a common factor of 4. Thus, we can factor it out, leaving us with: . We recognize that the expression inside the parentheses is a difference of squares, and factors as such: . Finally, we are done.

The common factor here is . Pull this out of both terms to simplify:

Factor:

When factoring a polynomial , the product of the coefficients must be , the sum of the factors must be , and the product of the factors must be .

For the above equation, , , and .

Set up the factor equation:

Becauase is negative, one of the factors must be negative as well. Because is positive, this means the larger factor is positive as well.

Two numbers that meet these requirements are and . Their product is , and their sum is .

To factor we need to find two numbers that add to get , and multiply to get . Because the has a negative sign, we can take the factors of , which are and and see which would subtract to get to . Due to the being positive, we know that the larger number in each factoring pair would have to also be positive. We can see that would not equal , but does, so:

In order to factor this, we will first need to pull a negative 1 out as the common factor.

Factor the term in the parentheses.

The factors of the first term are:

The factors of the last term are:

Using trial and error, we can determine that the factors for the term and for the number 14 will work since the numbers can be manipulated to achieve the middle term.

Remember to add the negative sign in front of the binomials.

The answer is:

The first thing we can do is factor a out of both terms:

Now we can see that each term is the square of something simpler:

We can use our squared terms formula to solve:

The easiest way to solve this problem is to factor the original function, and then to find the zeroes from the factored form. To do this, we start with the original function, f(x).

Next, we need to set up the function in factored form, leaving blanks for the numbers we don't yet know.

At this point, we need to find two numbers - one for each blank. By looking at the original function, we can gather a few clues that will help us find the two numbers. The product of these two numbers will be equal to the last term of our original function (-14, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of our original function (-5, or b in the standard quadratic formula). Because their product is negative (-14), that must mean that they have different signs - otherwise, their product would be positive. Also, because their sum is negative (-5), we know that the larger of the two numbers must be negative, otherwise their sum would be positive.

Now, at this point, we may need to test a few different possibilities, using the clues we gathered from the original function. In the end, we'll find that the only numbers that work here are 2 and -7, as the product of 2 and -7 is -14, and sum of 2 and -7 is -5. So, this results in our function's factored form looking like...

Now, the final step in this problem is finding the zeros. To do this, we need to think about what a zero is. A zero is the x-value(s) at which...

So, to solve for our zeros, we just need to set the right side of our function equal to zero and solve for x.

Because if either of these two factors is equal to zero, the entire function will be equal to zero (as anything multiplied by zero is zero), we can consider each of the two factors separately and solve for x. We'll start with the factor on the left.

We'll finish with the factor from the right.

Now, we have both of our zeros and the answer to our problem...