Factor 38x² + 67x + 15

Here we will show you how to factor the quadratic function 38x² + 67x + 15 using the box method. In other words, we will show you how to factor 38x squared plus 67x plus 15 (38x^2 + 67x + 15) 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 + 67x + 15, like this:

a = 38
b = 67
c = 15

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

5	10x	15
19x	38x²	57x
	2x	3

We put 38x² (a) in the bottom left square and 15 (c) in the top right square, like this:

5	10x	15
19x	38x²	57x
	2x	3

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 15 (a × c), and add together to equal 67 (b).

More specifically, 38 times 15 is 570. Therefore, we need to find the two numbers that multiply to equal 570, and add to equal 67.

? × ? = 570
? + ? = 67

After looking at this problem, we can see that the two numbers that multiply together to equal 570, and add together to equal 67, are 10 and 57, as illustrated here:

10 × 57 = 570
10 + 57 = 67

Now, we can fill in the last two squares in our box with 10x and 57x. Place 10x in the upper left square, and place 57x in the lower right square.

5	10x	15
19x	38x²	57x
	2x	3

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 10x and 15. The greatest common factor of 10x and 15 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	10x	15
19x	38x²	57x
	2x	3

Next, let’s look at the bottom row. We have the terms 38x² and 57x. The greatest common factor of 38x² and 57x is 19x. Therefore, we write 19x to the left of the bottom row. You can see it here in the color blue:

5	10x	15
19x	38x²	57x
	2x	3

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

38x² ÷ 19x = 2x

You can see this value colored in orange below:

5	10x	15
19x	38x²	57x
	2x	3

Next, we divide 57x by 19x (labeled in blue). This gives us 3.

57x ÷ 19x = 3

You can see this value colored in purple below:

5	10x	15
19x	38x²	57x
	2x	3

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 + 67x + 15. 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:

(19x + 5)(2x + 3)

That’s it! Now you know how to factor the equation 38x² + 67x + 15.

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