Factor 64x² + 92x + 19

Here we will show you how to factor the quadratic function 64x² + 92x + 19 using the box method. In other words, we will show you how to factor 64x squared plus 92x plus 19 (64x^2 + 92x + 19) 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 64x² + 92x + 19, like this:

a = 64
b = 92
c = 19

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

1	16x	19
4x	64x²	76x
	16x	19

We put 64x² (a) in the bottom left square and 19 (c) in the top right square, like this:

1	16x	19
4x	64x²	76x
	16x	19

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 64 times 19 (a × c), and add together to equal 92 (b).

More specifically, 64 times 19 is 1216. Therefore, we need to find the two numbers that multiply to equal 1216, and add to equal 92.

? × ? = 1216
? + ? = 92

After looking at this problem, we can see that the two numbers that multiply together to equal 1216, and add together to equal 92, are 16 and 76, as illustrated here:

16 × 76 = 1216
16 + 76 = 92

Now, we can fill in the last two squares in our box with 16x and 76x. Place 16x in the upper left square, and place 76x in the lower right square.

1	16x	19
4x	64x²	76x
	16x	19

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 16x and 19. The greatest common factor of 16x and 19 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	16x	19
4x	64x²	76x
	16x	19

Next, let’s look at the bottom row. We have the terms 64x² and 76x. The greatest common factor of 64x² and 76x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

1	16x	19
4x	64x²	76x
	16x	19

To find the values below the table, we first divide 64x² by 4x (labeled in blue). This gives us 16x.

64x² ÷ 4x = 16x

You can see this value colored in orange below:

1	16x	19
4x	64x²	76x
	16x	19

Next, we divide 76x by 4x (labeled in blue). This gives us 19.

76x ÷ 4x = 19

You can see this value colored in purple below:

1	16x	19
4x	64x²	76x
	16x	19

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 64x² + 92x + 19. 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:

(4x + 1)(16x + 19)

That’s it! Now you know how to factor the equation 64x² + 92x + 19.

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