Factor 14x² + 77x + 84

Here we will show you how to factor the quadratic function 14x² + 77x + 84 using the box method. In other words, we will show you how to factor 14x squared plus 77x plus 84 (14x^2 + 77x + 84) 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 + 77x + 84, like this:

a = 14
b = 77
c = 84

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

21	21x	84
14x	14x²	56x
	x	4

We put 14x² (a) in the bottom left square and 84 (c) in the top right square, like this:

21	21x	84
14x	14x²	56x
	x	4

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 84 (a × c), and add together to equal 77 (b).

More specifically, 14 times 84 is 1176. Therefore, we need to find the two numbers that multiply to equal 1176, and add to equal 77.

? × ? = 1176
? + ? = 77

After looking at this problem, we can see that the two numbers that multiply together to equal 1176, and add together to equal 77, are 21 and 56, as illustrated here:

21 × 56 = 1176
21 + 56 = 77

Now, we can fill in the last two squares in our box with 21x and 56x. Place 21x in the upper left square, and place 56x in the lower right square.

21	21x	84
14x	14x²	56x
	x	4

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 21x and 84. The greatest common factor of 21x and 84 is 21. Therefore, we write 21 to the left of the top row. You can see it here in the color green:

21	21x	84
14x	14x²	56x
	x	4

Next, let’s look at the bottom row. We have the terms 14x² and 56x. The greatest common factor of 14x² and 56x is 14x. Therefore, we write 14x to the left of the bottom row. You can see it here in the color blue:

21	21x	84
14x	14x²	56x
	x	4

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

14x² ÷ 14x = x

You can see this value colored in orange below:

21	21x	84
14x	14x²	56x
	x	4

Next, we divide 56x by 14x (labeled in blue). This gives us 4.

56x ÷ 14x = 4

You can see this value colored in purple below:

21	21x	84
14x	14x²	56x
	x	4

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 + 77x + 84. 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:

(14x + 21)(x + 4)

That’s it! Now you know how to factor the equation 14x² + 77x + 84.

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