Factor 20x² + 97x + 23

Here we will show you how to factor the quadratic function 20x² + 97x + 23 using the box method. In other words, we will show you how to factor 20x squared plus 97x plus 23 (20x^2 + 97x + 23) 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 + 23, like this:

a = 20
b = 97
c = 23

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

1	5x	23
4x	20x²	92x
	5x	23

We put 20x² (a) in the bottom left square and 23 (c) in the top right square, like this:

1	5x	23
4x	20x²	92x
	5x	23

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 23 (a × c), and add together to equal 97 (b).

More specifically, 20 times 23 is 460. Therefore, we need to find the two numbers that multiply to equal 460, and add to equal 97.

? × ? = 460
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 460, and add together to equal 97, are 5 and 92, as illustrated here:

5 × 92 = 460
5 + 92 = 97

Now, we can fill in the last two squares in our box with 5x and 92x. Place 5x in the upper left square, and place 92x in the lower right square.

1	5x	23
4x	20x²	92x
	5x	23

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 5x and 23. The greatest common factor of 5x and 23 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	5x	23
4x	20x²	92x
	5x	23

Next, let’s look at the bottom row. We have the terms 20x² and 92x. The greatest common factor of 20x² and 92x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

1	5x	23
4x	20x²	92x
	5x	23

To find the values below the table, we first divide 20x² by 4x (labeled in blue). This gives us 5x.

20x² ÷ 4x = 5x

You can see this value colored in orange below:

1	5x	23
4x	20x²	92x
	5x	23

Next, we divide 92x by 4x (labeled in blue). This gives us 23.

92x ÷ 4x = 23

You can see this value colored in purple below:

1	5x	23
4x	20x²	92x
	5x	23

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 + 23. 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)(5x + 23)

That’s it! Now you know how to factor the equation 20x² + 97x + 23.

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