Factor 14x² + 89x + 75

Here we will show you how to factor the quadratic function 14x² + 89x + 75 using the box method. In other words, we will show you how to factor 14x squared plus 89x plus 75 (14x^2 + 89x + 75) 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 + 89x + 75, like this:

a = 14
b = 89
c = 75

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

1	14x	75
x	14x²	75x
	14x	75

We put 14x² (a) in the bottom left square and 75 (c) in the top right square, like this:

1	14x	75
x	14x²	75x
	14x	75

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 75 (a × c), and add together to equal 89 (b).

More specifically, 14 times 75 is 1050. Therefore, we need to find the two numbers that multiply to equal 1050, and add to equal 89.

? × ? = 1050
? + ? = 89

After looking at this problem, we can see that the two numbers that multiply together to equal 1050, and add together to equal 89, are 14 and 75, as illustrated here:

14 × 75 = 1050
14 + 75 = 89

Now, we can fill in the last two squares in our box with 14x and 75x. Place 14x in the upper left square, and place 75x in the lower right square.

1	14x	75
x	14x²	75x
	14x	75

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 75. The greatest common factor of 14x and 75 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	14x	75
x	14x²	75x
	14x	75

Next, let’s look at the bottom row. We have the terms 14x² and 75x. The greatest common factor of 14x² and 75x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	14x	75
x	14x²	75x
	14x	75

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	75
x	14x²	75x
	14x	75

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

75x ÷ x = 75

You can see this value colored in purple below:

1	14x	75
x	14x²	75x
	14x	75

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 + 89x + 75. 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 + 75)

That’s it! Now you know how to factor the equation 14x² + 89x + 75.

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