Factor 31x² + 73x + 22

Here we will show you how to factor the quadratic function 31x² + 73x + 22 using the box method. In other words, we will show you how to factor 31x squared plus 73x plus 22 (31x^2 + 73x + 22) 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 31x² + 73x + 22, like this:

a = 31
b = 73
c = 22

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

11	11x	22
31x	31x²	62x
	x	2

We put 31x² (a) in the bottom left square and 22 (c) in the top right square, like this:

11	11x	22
31x	31x²	62x
	x	2

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 31 times 22 (a × c), and add together to equal 73 (b).

More specifically, 31 times 22 is 682. Therefore, we need to find the two numbers that multiply to equal 682, and add to equal 73.

? × ? = 682
? + ? = 73

After looking at this problem, we can see that the two numbers that multiply together to equal 682, and add together to equal 73, are 11 and 62, as illustrated here:

11 × 62 = 682
11 + 62 = 73

Now, we can fill in the last two squares in our box with 11x and 62x. Place 11x in the upper left square, and place 62x in the lower right square.

11	11x	22
31x	31x²	62x
	x	2

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 11x and 22. The greatest common factor of 11x and 22 is 11. Therefore, we write 11 to the left of the top row. You can see it here in the color green:

11	11x	22
31x	31x²	62x
	x	2

Next, let’s look at the bottom row. We have the terms 31x² and 62x. The greatest common factor of 31x² and 62x is 31x. Therefore, we write 31x to the left of the bottom row. You can see it here in the color blue:

11	11x	22
31x	31x²	62x
	x	2

To find the values below the table, we first divide 31x² by 31x (labeled in blue). This gives us x.

31x² ÷ 31x = x

You can see this value colored in orange below:

11	11x	22
31x	31x²	62x
	x	2

Next, we divide 62x by 31x (labeled in blue). This gives us 2.

62x ÷ 31x = 2

You can see this value colored in purple below:

11	11x	22
31x	31x²	62x
	x	2

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 31x² + 73x + 22. 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:

(31x + 11)(x + 2)

That’s it! Now you know how to factor the equation 31x² + 73x + 22.

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