Factor 28x² + 41x + 13

Here we will show you how to factor the quadratic function 28x² + 41x + 13 using the box method. In other words, we will show you how to factor 28x squared plus 41x plus 13 (28x^2 + 41x + 13) 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 28x² + 41x + 13, like this:

a = 28
b = 41
c = 13

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

13	13x	13
28x	28x²	28x
	x	1

We put 28x² (a) in the bottom left square and 13 (c) in the top right square, like this:

13	13x	13
28x	28x²	28x
	x	1

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 28 times 13 (a × c), and add together to equal 41 (b).

More specifically, 28 times 13 is 364. Therefore, we need to find the two numbers that multiply to equal 364, and add to equal 41.

? × ? = 364
? + ? = 41

After looking at this problem, we can see that the two numbers that multiply together to equal 364, and add together to equal 41, are 13 and 28, as illustrated here:

13 × 28 = 364
13 + 28 = 41

Now, we can fill in the last two squares in our box with 13x and 28x. Place 13x in the upper left square, and place 28x in the lower right square.

13	13x	13
28x	28x²	28x
	x	1

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 13x and 13. The greatest common factor of 13x and 13 is 13. Therefore, we write 13 to the left of the top row. You can see it here in the color green:

13	13x	13
28x	28x²	28x
	x	1

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

13	13x	13
28x	28x²	28x
	x	1

To find the values below the table, we first divide 28x² by 28x (labeled in blue). This gives us x.

28x² ÷ 28x = x

You can see this value colored in orange below:

13	13x	13
28x	28x²	28x
	x	1

Next, we divide 28x by 28x (labeled in blue). This gives us 1.

28x ÷ 28x = 1

You can see this value colored in purple below:

13	13x	13
28x	28x²	28x
	x	1

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 28x² + 41x + 13. 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:

(28x + 13)(x + 1)

That’s it! Now you know how to factor the equation 28x² + 41x + 13.

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