Factor 3x² + 34x + 56

Here we will show you how to factor the quadratic function 3x² + 34x + 56 using the box method. In other words, we will show you how to factor 3x squared plus 34x plus 56 (3x^2 + 34x + 56) 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 3x² + 34x + 56, like this:

a = 3
b = 34
c = 56

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

2	6x	56
x	3x²	28x
	3x	28

We put 3x² (a) in the bottom left square and 56 (c) in the top right square, like this:

2	6x	56
x	3x²	28x
	3x	28

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 3 times 56 (a × c), and add together to equal 34 (b).

More specifically, 3 times 56 is 168. Therefore, we need to find the two numbers that multiply to equal 168, and add to equal 34.

? × ? = 168
? + ? = 34

After looking at this problem, we can see that the two numbers that multiply together to equal 168, and add together to equal 34, are 6 and 28, as illustrated here:

6 × 28 = 168
6 + 28 = 34

Now, we can fill in the last two squares in our box with 6x and 28x. Place 6x in the upper left square, and place 28x in the lower right square.

2	6x	56
x	3x²	28x
	3x	28

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 6x and 56. The greatest common factor of 6x and 56 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	6x	56
x	3x²	28x
	3x	28

Next, let’s look at the bottom row. We have the terms 3x² and 28x. The greatest common factor of 3x² and 28x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

2	6x	56
x	3x²	28x
	3x	28

To find the values below the table, we first divide 3x² by x (labeled in blue). This gives us 3x.

3x² ÷ x = 3x

You can see this value colored in orange below:

2	6x	56
x	3x²	28x
	3x	28

Next, we divide 28x by x (labeled in blue). This gives us 28.

28x ÷ x = 28

You can see this value colored in purple below:

2	6x	56
x	3x²	28x
	3x	28

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 3x² + 34x + 56. 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 + 2)(3x + 28)

That’s it! Now you know how to factor the equation 3x² + 34x + 56.

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