Factor 12x² + 56x + 49

Here we will show you how to factor the quadratic function 12x² + 56x + 49 using the box method. In other words, we will show you how to factor 12x squared plus 56x plus 49 (12x^2 + 56x + 49) 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 12x² + 56x + 49, like this:

a = 12
b = 56
c = 49

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

7	14x	49
6x	12x²	42x
	2x	7

We put 12x² (a) in the bottom left square and 49 (c) in the top right square, like this:

7	14x	49
6x	12x²	42x
	2x	7

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 12 times 49 (a × c), and add together to equal 56 (b).

More specifically, 12 times 49 is 588. Therefore, we need to find the two numbers that multiply to equal 588, and add to equal 56.

? × ? = 588
? + ? = 56

After looking at this problem, we can see that the two numbers that multiply together to equal 588, and add together to equal 56, are 14 and 42, as illustrated here:

14 × 42 = 588
14 + 42 = 56

Now, we can fill in the last two squares in our box with 14x and 42x. Place 14x in the upper left square, and place 42x in the lower right square.

7	14x	49
6x	12x²	42x
	2x	7

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 49. The greatest common factor of 14x and 49 is 7. Therefore, we write 7 to the left of the top row. You can see it here in the color green:

7	14x	49
6x	12x²	42x
	2x	7

Next, let’s look at the bottom row. We have the terms 12x² and 42x. The greatest common factor of 12x² and 42x is 6x. Therefore, we write 6x to the left of the bottom row. You can see it here in the color blue:

7	14x	49
6x	12x²	42x
	2x	7

To find the values below the table, we first divide 12x² by 6x (labeled in blue). This gives us 2x.

12x² ÷ 6x = 2x

You can see this value colored in orange below:

7	14x	49
6x	12x²	42x
	2x	7

Next, we divide 42x by 6x (labeled in blue). This gives us 7.

42x ÷ 6x = 7

You can see this value colored in purple below:

7	14x	49
6x	12x²	42x
	2x	7

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 12x² + 56x + 49. 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:

(6x + 7)(2x + 7)

That’s it! Now you know how to factor the equation 12x² + 56x + 49.

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