Factor 20x² + 97x + 51

Here we will show you how to factor the quadratic function 20x² + 97x + 51 using the box method. In other words, we will show you how to factor 20x squared plus 97x plus 51 (20x^2 + 97x + 51) 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 + 51, like this:

a = 20
b = 97
c = 51

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

3	12x	51
5x	20x²	85x
	4x	17

We put 20x² (a) in the bottom left square and 51 (c) in the top right square, like this:

3	12x	51
5x	20x²	85x
	4x	17

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

More specifically, 20 times 51 is 1020. Therefore, we need to find the two numbers that multiply to equal 1020, and add to equal 97.

? × ? = 1020
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 1020, and add together to equal 97, are 12 and 85, as illustrated here:

12 × 85 = 1020
12 + 85 = 97

Now, we can fill in the last two squares in our box with 12x and 85x. Place 12x in the upper left square, and place 85x in the lower right square.

3	12x	51
5x	20x²	85x
	4x	17

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 12x and 51. The greatest common factor of 12x and 51 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	12x	51
5x	20x²	85x
	4x	17

Next, let’s look at the bottom row. We have the terms 20x² and 85x. The greatest common factor of 20x² and 85x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

3	12x	51
5x	20x²	85x
	4x	17

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

20x² ÷ 5x = 4x

You can see this value colored in orange below:

3	12x	51
5x	20x²	85x
	4x	17

Next, we divide 85x by 5x (labeled in blue). This gives us 17.

85x ÷ 5x = 17

You can see this value colored in purple below:

3	12x	51
5x	20x²	85x
	4x	17

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 + 51. 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:

(5x + 3)(4x + 17)

That’s it! Now you know how to factor the equation 20x² + 97x + 51.

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