Factor 12x² - 67x + 65

Here we will show you how to factor the quadratic function 12x² - 67x + 65 using the box method. In other words, we will show you how to factor 12x squared minus 67x plus 65 (12x^2 - 67x + 65) 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 - 67x + 65, like this:

a = 12
b = -67
c = 65

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

-13	-52x	65
3x	12x²	-15x
	4x	-5

We put 12x² (a) in the bottom left square and 65 (c) in the top right square, like this:

-13	-52x	65
3x	12x²	-15x
	4x	-5

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 65 (a × c), and add together to equal -67 (b).

More specifically, 12 times 65 is 780. Therefore, we need to find the two numbers that multiply to equal 780, and add to equal -67.

? × ? = 780
? + ? = -67

After looking at this problem, we can see that the two numbers that multiply together to equal 780, and add together to equal -67, are -52 and -15, as illustrated here:

-52 × -15 = 780
-52 + -15 = -67

Now, we can fill in the last two squares in our box with -52x and -15x. Place -52x in the upper left square, and place -15x in the lower right square.

-13	-52x	65
3x	12x²	-15x
	4x	-5

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 -52x and 65. The greatest common factor of -52x and 65 is -13. Therefore, we write -13 to the left of the top row. You can see it here in the color green:

-13	-52x	65
3x	12x²	-15x
	4x	-5

Next, let’s look at the bottom row. We have the terms 12x² and -15x. The greatest common factor of 12x² and -15x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

-13	-52x	65
3x	12x²	-15x
	4x	-5

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:

-13	-52x	65
3x	12x²	-15x
	4x	-5

Next, we divide -15x by 3x (labeled in blue). This gives us -5.

-15x ÷ 3x = -5

You can see this value colored in purple below:

-13	-52x	65
3x	12x²	-15x
	4x	-5

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 - 67x + 65. 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 - 13)(4x - 5)

That’s it! Now you know how to factor the equation 12x² - 67x + 65.

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