You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

,

where and .

To find , you use the formula for perfect squares:

,

where and .

Substituting above, the final answer is .

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:

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.

You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

,

where and .

To find , you use the formula for perfect squares:

,

where and .

Substituting above, the final answer is .

To find , we must start inwards and work our way outwards, i.e. starting with :

We can now use this value to find as follows:

Our final answer is therefore

When finding the equation of a quadratic from its zeros, the natural first step is to recreate the factored form of a simple quadratic with those zeros. Putting aside the coefficient, a quadratic with zeros at 3 and -6 would factor to:

So you know that a possible quadratic for these zeros would be:

Now you need to determine the coefficient of the quadratic, and that’s where the point (2, -24) comes in. That means that if you plug in x = 2, the result of the quadratic will be -24. So you can set up the equation:

, where a is the coefficient you’re solving for. And you know that this is true when x = 2, so if you plug in x = 2 you can solve for a:

So a = 3. When you then distribute that coefficient of 3 across the original simple quadratic, you have:

So your quadratic is

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.

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_)