Factor 16x² + 77x + 61

Here we will show you how to factor the quadratic function 16x² + 77x + 61 using the box method. In other words, we will show you how to factor 16x squared plus 77x plus 61 (16x^2 + 77x + 61) 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 16x² + 77x + 61, like this:

a = 16
b = 77
c = 61

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

1	16x	61
x	16x²	61x
	16x	61

We put 16x² (a) in the bottom left square and 61 (c) in the top right square, like this:

1	16x	61
x	16x²	61x
	16x	61

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 16 times 61 (a × c), and add together to equal 77 (b).

More specifically, 16 times 61 is 976. Therefore, we need to find the two numbers that multiply to equal 976, and add to equal 77.

? × ? = 976
? + ? = 77

After looking at this problem, we can see that the two numbers that multiply together to equal 976, and add together to equal 77, are 16 and 61, as illustrated here:

16 × 61 = 976
16 + 61 = 77

Now, we can fill in the last two squares in our box with 16x and 61x. Place 16x in the upper left square, and place 61x in the lower right square.

1	16x	61
x	16x²	61x
	16x	61

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 61. The greatest common factor of 16x and 61 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	16x	61
x	16x²	61x
	16x	61

Next, let’s look at the bottom row. We have the terms 16x² and 61x. The greatest common factor of 16x² and 61x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	16x	61
x	16x²	61x
	16x	61

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

16x² ÷ x = 16x

You can see this value colored in orange below:

1	16x	61
x	16x²	61x
	16x	61

Next, we divide 61x by x (labeled in blue). This gives us 61.

61x ÷ x = 61

You can see this value colored in purple below:

1	16x	61
x	16x²	61x
	16x	61

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 16x² + 77x + 61. 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:

(x + 1)(16x + 61)

That’s it! Now you know how to factor the equation 16x² + 77x + 61.

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