Factor 25x² + 97x + 66

Here we will show you how to factor the quadratic function 25x² + 97x + 66 using the box method. In other words, we will show you how to factor 25x squared plus 97x plus 66 (25x^2 + 97x + 66) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 25x² + 97x + 66, like this:

a = 25
b = 97
c = 66

Step 2: Next, we need to draw a box and divide it into four squares:

22	22x	66
25x	25x²	75x
	x	3

We put 25x² (a) in the bottom left square and 66 (c) in the top right square, like this:

22	22x	66
25x	25x²	75x
	x	3

Step 3: In order to fill in the other two squares, we need to do some math. We have to find two numbers that multiply together to give us the product of 25 times 66 (a × c), and add together to equal 97 (b).

More specifically, 25 times 66 is 1650. Therefore, we need to find the two numbers that multiply to equal 1650, and add to equal 97.

? × ? = 1650
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 1650, and add together to equal 97, are 22 and 75, as illustrated here:

22 × 75 = 1650
22 + 75 = 97

Now, we can fill in the last two squares in our box with 22x and 75x. Place 22x in the upper left square, and place 75x in the lower right square.

22	22x	66
25x	25x²	75x
	x	3

Step 4: Next, we need to find four final numbers to finish factoring our equation. We can see that our box is a 2 x 2 table made up of rows and columns.

Our goal is to find the numbers that, when multiplied together, result in the products in each square. We can do this by finding the greatest common factor in each row.

Let’s look at the top row. We have the terms 22x and 66. The greatest common factor of 22x and 66 is 22. Therefore, we write 22 to the left of the top row. You can see it here in the color green:

22	22x	66
25x	25x²	75x
	x	3

Next, let’s look at the bottom row. We have the terms 25x² and 75x. The greatest common factor of 25x² and 75x is 25x. Therefore, we write 25x to the left of the bottom row. You can see it here in the color blue:

22	22x	66
25x	25x²	75x
	x	3

To find the values below the table, we first divide 25x² by 25x (labeled in blue). This gives us x.

25x² ÷ 25x = x

You can see this value colored in orange below:

22	22x	66
25x	25x²	75x
	x	3

Next, we divide 75x by 25x (labeled in blue). This gives us 3.

75x ÷ 25x = 3

You can see this value colored in purple below:

22	22x	66
25x	25x²	75x
	x	3

Step 5: Now we have all of the information we need to finish factoring the equation. The values outside of the box are used to factor 25x² + 97x + 66. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(25x + 22)(x + 3)

That’s it! Now you know how to factor the equation 25x² + 97x + 66.

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