Factor 12x² + 55x + 63

Here we will show you how to factor the quadratic function 12x² + 55x + 63 using the box method. In other words, we will show you how to factor 12x squared plus 55x plus 63 (12x^2 + 55x + 63) 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 + 55x + 63, like this:

a = 12
b = 55
c = 63

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

9	27x	63
4x	12x²	28x
	3x	7

We put 12x² (a) in the bottom left square and 63 (c) in the top right square, like this:

9	27x	63
4x	12x²	28x
	3x	7

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 63 (a × c), and add together to equal 55 (b).

More specifically, 12 times 63 is 756. Therefore, we need to find the two numbers that multiply to equal 756, and add to equal 55.

? × ? = 756
? + ? = 55

After looking at this problem, we can see that the two numbers that multiply together to equal 756, and add together to equal 55, are 27 and 28, as illustrated here:

27 × 28 = 756
27 + 28 = 55

Now, we can fill in the last two squares in our box with 27x and 28x. Place 27x in the upper left square, and place 28x in the lower right square.

9	27x	63
4x	12x²	28x
	3x	7

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 27x and 63. The greatest common factor of 27x and 63 is 9. Therefore, we write 9 to the left of the top row. You can see it here in the color green:

9	27x	63
4x	12x²	28x
	3x	7

Next, let’s look at the bottom row. We have the terms 12x² and 28x. The greatest common factor of 12x² and 28x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

9	27x	63
4x	12x²	28x
	3x	7

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

12x² ÷ 4x = 3x

You can see this value colored in orange below:

9	27x	63
4x	12x²	28x
	3x	7

Next, we divide 28x by 4x (labeled in blue). This gives us 7.

28x ÷ 4x = 7

You can see this value colored in purple below:

9	27x	63
4x	12x²	28x
	3x	7

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 + 55x + 63. 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 + 9)(3x + 7)

That’s it! Now you know how to factor the equation 12x² + 55x + 63.

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