Factor 12x² + 63x + 51

Here we will show you how to factor the quadratic function 12x² + 63x + 51 using the box method. In other words, we will show you how to factor 12x squared plus 63x plus 51 (12x^2 + 63x + 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 12x² + 63x + 51, like this:

a = 12
b = 63
c = 51

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

3	12x	51
3x	12x²	51x
	4x	17

We put 12x² (a) in the bottom left square and 51 (c) in the top right square, like this:

3	12x	51
3x	12x²	51x
	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 12 times 51 (a × c), and add together to equal 63 (b).

More specifically, 12 times 51 is 612. Therefore, we need to find the two numbers that multiply to equal 612, and add to equal 63.

? × ? = 612
? + ? = 63

After looking at this problem, we can see that the two numbers that multiply together to equal 612, and add together to equal 63, are 12 and 51, as illustrated here:

12 × 51 = 612
12 + 51 = 63

Now, we can fill in the last two squares in our box with 12x and 51x. Place 12x in the upper left square, and place 51x in the lower right square.

3	12x	51
3x	12x²	51x
	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
3x	12x²	51x
	4x	17

Next, let’s look at the bottom row. We have the terms 12x² and 51x. The greatest common factor of 12x² and 51x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

3	12x	51
3x	12x²	51x
	4x	17

To find the values below the table, we first divide 12x² by 3x (labeled in blue). This gives us 4x.

12x² ÷ 3x = 4x

You can see this value colored in orange below:

3	12x	51
3x	12x²	51x
	4x	17

Next, we divide 51x by 3x (labeled in blue). This gives us 17.

51x ÷ 3x = 17

You can see this value colored in purple below:

3	12x	51
3x	12x²	51x
	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 12x² + 63x + 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:

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

That’s it! Now you know how to factor the equation 12x² + 63x + 51.

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