Factor 20x² + 97x + 77

Here we will show you how to factor the quadratic function 20x² + 97x + 77 using the box method. In other words, we will show you how to factor 20x squared plus 97x plus 77 (20x^2 + 97x + 77) 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 20x² + 97x + 77, like this:

a = 20
b = 97
c = 77

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

1	20x	77
x	20x²	77x
	20x	77

We put 20x² (a) in the bottom left square and 77 (c) in the top right square, like this:

1	20x	77
x	20x²	77x
	20x	77

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 20 times 77 (a × c), and add together to equal 97 (b).

More specifically, 20 times 77 is 1540. Therefore, we need to find the two numbers that multiply to equal 1540, and add to equal 97.

? × ? = 1540
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 1540, and add together to equal 97, are 20 and 77, as illustrated here:

20 × 77 = 1540
20 + 77 = 97

Now, we can fill in the last two squares in our box with 20x and 77x. Place 20x in the upper left square, and place 77x in the lower right square.

1	20x	77
x	20x²	77x
	20x	77

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 20x and 77. The greatest common factor of 20x and 77 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	20x	77
x	20x²	77x
	20x	77

Next, let’s look at the bottom row. We have the terms 20x² and 77x. The greatest common factor of 20x² and 77x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	20x	77
x	20x²	77x
	20x	77

To find the values below the table, we first divide 20x² by x (labeled in blue). This gives us 20x.

20x² ÷ x = 20x

You can see this value colored in orange below:

1	20x	77
x	20x²	77x
	20x	77

Next, we divide 77x by x (labeled in blue). This gives us 77.

77x ÷ x = 77

You can see this value colored in purple below:

1	20x	77
x	20x²	77x
	20x	77

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 20x² + 97x + 77. 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)(20x + 77)

That’s it! Now you know how to factor the equation 20x² + 97x + 77.

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