Factor 38x² + 53x + 17

Here we will show you how to factor the quadratic function 38x² + 53x + 17 using the box method. In other words, we will show you how to factor 38x squared plus 53x plus 17 (38x^2 + 53x + 17) 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 38x² + 53x + 17, like this:

a = 38
b = 53
c = 17

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

1	19x	17
2x	38x²	34x
	19x	17

We put 38x² (a) in the bottom left square and 17 (c) in the top right square, like this:

1	19x	17
2x	38x²	34x
	19x	17

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 38 times 17 (a × c), and add together to equal 53 (b).

More specifically, 38 times 17 is 646. Therefore, we need to find the two numbers that multiply to equal 646, and add to equal 53.

? × ? = 646
? + ? = 53

After looking at this problem, we can see that the two numbers that multiply together to equal 646, and add together to equal 53, are 19 and 34, as illustrated here:

19 × 34 = 646
19 + 34 = 53

Now, we can fill in the last two squares in our box with 19x and 34x. Place 19x in the upper left square, and place 34x in the lower right square.

1	19x	17
2x	38x²	34x
	19x	17

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 19x and 17. The greatest common factor of 19x and 17 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	19x	17
2x	38x²	34x
	19x	17

Next, let’s look at the bottom row. We have the terms 38x² and 34x. The greatest common factor of 38x² and 34x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	19x	17
2x	38x²	34x
	19x	17

To find the values below the table, we first divide 38x² by 2x (labeled in blue). This gives us 19x.

38x² ÷ 2x = 19x

You can see this value colored in orange below:

1	19x	17
2x	38x²	34x
	19x	17

Next, we divide 34x by 2x (labeled in blue). This gives us 17.

34x ÷ 2x = 17

You can see this value colored in purple below:

1	19x	17
2x	38x²	34x
	19x	17

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 38x² + 53x + 17. 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:

(2x + 1)(19x + 17)

That’s it! Now you know how to factor the equation 38x² + 53x + 17.

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