Factor 12x² + 62x + 78

Here we will show you how to factor the quadratic function 12x² + 62x + 78 using the box method. In other words, we will show you how to factor 12x squared plus 62x plus 78 (12x^2 + 62x + 78) 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 + 62x + 78, like this:

a = 12
b = 62
c = 78

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

26	26x	78
12x	12x²	36x
	x	3

We put 12x² (a) in the bottom left square and 78 (c) in the top right square, like this:

26	26x	78
12x	12x²	36x
	x	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 12 times 78 (a × c), and add together to equal 62 (b).

More specifically, 12 times 78 is 936. Therefore, we need to find the two numbers that multiply to equal 936, and add to equal 62.

? × ? = 936
? + ? = 62

After looking at this problem, we can see that the two numbers that multiply together to equal 936, and add together to equal 62, are 26 and 36, as illustrated here:

26 × 36 = 936
26 + 36 = 62

Now, we can fill in the last two squares in our box with 26x and 36x. Place 26x in the upper left square, and place 36x in the lower right square.

26	26x	78
12x	12x²	36x
	x	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 26x and 78. The greatest common factor of 26x and 78 is 26. Therefore, we write 26 to the left of the top row. You can see it here in the color green:

26	26x	78
12x	12x²	36x
	x	3

Next, let’s look at the bottom row. We have the terms 12x² and 36x. The greatest common factor of 12x² and 36x is 12x. Therefore, we write 12x to the left of the bottom row. You can see it here in the color blue:

26	26x	78
12x	12x²	36x
	x	3

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

12x² ÷ 12x = x

You can see this value colored in orange below:

26	26x	78
12x	12x²	36x
	x	3

Next, we divide 36x by 12x (labeled in blue). This gives us 3.

36x ÷ 12x = 3

You can see this value colored in purple below:

26	26x	78
12x	12x²	36x
	x	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 12x² + 62x + 78. 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:

(12x + 26)(x + 3)

That’s it! Now you know how to factor the equation 12x² + 62x + 78.

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