Factor 13x² + 27x + 14

Here we will show you how to factor the quadratic function 13x² + 27x + 14 using the box method. In other words, we will show you how to factor 13x squared plus 27x plus 14 (13x^2 + 27x + 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 13x² + 27x + 14, like this:

a = 13
b = 27
c = 14

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

1	13x	14
x	13x²	14x
	13x	14

We put 13x² (a) in the bottom left square and 14 (c) in the top right square, like this:

1	13x	14
x	13x²	14x
	13x	14

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 13 times 14 (a × c), and add together to equal 27 (b).

More specifically, 13 times 14 is 182. Therefore, we need to find the two numbers that multiply to equal 182, and add to equal 27.

? × ? = 182
? + ? = 27

After looking at this problem, we can see that the two numbers that multiply together to equal 182, and add together to equal 27, are 13 and 14, as illustrated here:

13 × 14 = 182
13 + 14 = 27

Now, we can fill in the last two squares in our box with 13x and 14x. Place 13x in the upper left square, and place 14x in the lower right square.

1	13x	14
x	13x²	14x
	13x	14

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 13x and 14. The greatest common factor of 13x and 14 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	13x	14
x	13x²	14x
	13x	14

Next, let’s look at the bottom row. We have the terms 13x² and 14x. The greatest common factor of 13x² and 14x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	13x	14
x	13x²	14x
	13x	14

To find the values below the table, we first divide 13x² by x (labeled in blue). This gives us 13x.

13x² ÷ x = 13x

You can see this value colored in orange below:

1	13x	14
x	13x²	14x
	13x	14

Next, we divide 14x by x (labeled in blue). This gives us 14.

14x ÷ x = 14

You can see this value colored in purple below:

1	13x	14
x	13x²	14x
	13x	14

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 13x² + 27x + 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:

(x + 1)(13x + 14)

That’s it! Now you know how to factor the equation 13x² + 27x + 14.

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