Factor 37x² + 51x + 14

Here we will show you how to factor the quadratic function 37x² + 51x + 14 using the box method. In other words, we will show you how to factor 37x squared plus 51x plus 14 (37x^2 + 51x + 14) 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 37x² + 51x + 14, like this:

a = 37
b = 51
c = 14

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

14	14x	14
37x	37x²	37x
	x	1

We put 37x² (a) in the bottom left square and 14 (c) in the top right square, like this:

14	14x	14
37x	37x²	37x
	x	1

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 37 times 14 (a × c), and add together to equal 51 (b).

More specifically, 37 times 14 is 518. Therefore, we need to find the two numbers that multiply to equal 518, and add to equal 51.

? × ? = 518
? + ? = 51

After looking at this problem, we can see that the two numbers that multiply together to equal 518, and add together to equal 51, are 14 and 37, as illustrated here:

14 × 37 = 518
14 + 37 = 51

Now, we can fill in the last two squares in our box with 14x and 37x. Place 14x in the upper left square, and place 37x in the lower right square.

14	14x	14
37x	37x²	37x
	x	1

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 14x and 14. The greatest common factor of 14x and 14 is 14. Therefore, we write 14 to the left of the top row. You can see it here in the color green:

14	14x	14
37x	37x²	37x
	x	1

Next, let’s look at the bottom row. We have the terms 37x² and 37x. The greatest common factor of 37x² and 37x is 37x. Therefore, we write 37x to the left of the bottom row. You can see it here in the color blue:

14	14x	14
37x	37x²	37x
	x	1

To find the values below the table, we first divide 37x² by 37x (labeled in blue). This gives us x.

37x² ÷ 37x = x

You can see this value colored in orange below:

14	14x	14
37x	37x²	37x
	x	1

Next, we divide 37x by 37x (labeled in blue). This gives us 1.

37x ÷ 37x = 1

You can see this value colored in purple below:

14	14x	14
37x	37x²	37x
	x	1

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 37x² + 51x + 14. 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:

(37x + 14)(x + 1)

That’s it! Now you know how to factor the equation 37x² + 51x + 14.

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