Factor 14x² + 73x + 59

Here we will show you how to factor the quadratic function 14x² + 73x + 59 using the box method. In other words, we will show you how to factor 14x squared plus 73x plus 59 (14x^2 + 73x + 59) 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 14x² + 73x + 59, like this:

a = 14
b = 73
c = 59

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

1	14x	59
x	14x²	59x
	14x	59

We put 14x² (a) in the bottom left square and 59 (c) in the top right square, like this:

1	14x	59
x	14x²	59x
	14x	59

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 14 times 59 (a × c), and add together to equal 73 (b).

More specifically, 14 times 59 is 826. Therefore, we need to find the two numbers that multiply to equal 826, and add to equal 73.

? × ? = 826
? + ? = 73

After looking at this problem, we can see that the two numbers that multiply together to equal 826, and add together to equal 73, are 14 and 59, as illustrated here:

14 × 59 = 826
14 + 59 = 73

Now, we can fill in the last two squares in our box with 14x and 59x. Place 14x in the upper left square, and place 59x in the lower right square.

1	14x	59
x	14x²	59x
	14x	59

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 59. The greatest common factor of 14x and 59 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	14x	59
x	14x²	59x
	14x	59

Next, let’s look at the bottom row. We have the terms 14x² and 59x. The greatest common factor of 14x² and 59x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	14x	59
x	14x²	59x
	14x	59

To find the values below the table, we first divide 14x² by x (labeled in blue). This gives us 14x.

14x² ÷ x = 14x

You can see this value colored in orange below:

1	14x	59
x	14x²	59x
	14x	59

Next, we divide 59x by x (labeled in blue). This gives us 59.

59x ÷ x = 59

You can see this value colored in purple below:

1	14x	59
x	14x²	59x
	14x	59

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 14x² + 73x + 59. 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)(14x + 59)

That’s it! Now you know how to factor the equation 14x² + 73x + 59.

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