Factor 3x² + 34x + 31

Here we will show you how to factor the quadratic function 3x² + 34x + 31 using the box method. In other words, we will show you how to factor 3x squared plus 34x plus 31 (3x^2 + 34x + 31) 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 + 31, like this:

a = 3
b = 34
c = 31

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

1	3x	31
x	3x²	31x
	3x	31

We put 3x² (a) in the bottom left square and 31 (c) in the top right square, like this:

1	3x	31
x	3x²	31x
	3x	31

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 31 (a × c), and add together to equal 34 (b).

More specifically, 3 times 31 is 93. Therefore, we need to find the two numbers that multiply to equal 93, and add to equal 34.

? × ? = 93
? + ? = 34

After looking at this problem, we can see that the two numbers that multiply together to equal 93, and add together to equal 34, are 3 and 31, as illustrated here:

3 × 31 = 93
3 + 31 = 34

Now, we can fill in the last two squares in our box with 3x and 31x. Place 3x in the upper left square, and place 31x in the lower right square.

1	3x	31
x	3x²	31x
	3x	31

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 3x and 31. The greatest common factor of 3x and 31 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	3x	31
x	3x²	31x
	3x	31

Next, let’s look at the bottom row. We have the terms 3x² and 31x. The greatest common factor of 3x² and 31x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	3x	31
x	3x²	31x
	3x	31

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:

1	3x	31
x	3x²	31x
	3x	31

Next, we divide 31x by x (labeled in blue). This gives us 31.

31x ÷ x = 31

You can see this value colored in purple below:

1	3x	31
x	3x²	31x
	3x	31

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 + 31. 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)(3x + 31)

That’s it! Now you know how to factor the equation 3x² + 34x + 31.

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