Factor 68x² + 63x + 9

Here we will show you how to factor the quadratic function 68x² + 63x + 9 using the box method. In other words, we will show you how to factor 68x squared plus 63x plus 9 (68x^2 + 63x + 9) 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 68x² + 63x + 9, like this:

a = 68
b = 63
c = 9

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

3	12x	9
17x	68x²	51x
	4x	3

We put 68x² (a) in the bottom left square and 9 (c) in the top right square, like this:

3	12x	9
17x	68x²	51x
	4x	3

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 68 times 9 (a × c), and add together to equal 63 (b).

More specifically, 68 times 9 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	9
17x	68x²	51x
	4x	3

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 9. The greatest common factor of 12x and 9 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	12x	9
17x	68x²	51x
	4x	3

Next, let’s look at the bottom row. We have the terms 68x² and 51x. The greatest common factor of 68x² and 51x is 17x. Therefore, we write 17x to the left of the bottom row. You can see it here in the color blue:

3	12x	9
17x	68x²	51x
	4x	3

To find the values below the table, we first divide 68x² by 17x (labeled in blue). This gives us 4x.

68x² ÷ 17x = 4x

You can see this value colored in orange below:

3	12x	9
17x	68x²	51x
	4x	3

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

51x ÷ 17x = 3

You can see this value colored in purple below:

3	12x	9
17x	68x²	51x
	4x	3

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 68x² + 63x + 9. 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:

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

That’s it! Now you know how to factor the equation 68x² + 63x + 9.

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