Factor 48x² - 20x

Factor 48x² - 20x - 42

Here we will show you how to factor the quadratic function 48x² - 20x - 42 using the box method. In other words, we will show you how to factor 48x squared minus 20x minus 42 (48x^2 - 20x - 42) 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 48x² - 20x - 42, like this:

a = 48
b = -20
c = -42

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

-14	-56x	-42
12x	48x²	36x
	4x	3

We put 48x² (a) in the bottom left square and -42 (c) in the top right square, like this:

-14	-56x	-42
12x	48x²	36x
	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 48 times -42 (a × c), and add together to equal -20 (b).

More specifically, 48 times -42 is -2016. Therefore, we need to find the two numbers that multiply to equal -2016, and add to equal -20.

? × ? = -2016
? + ? = -20

After looking at this problem, we can see that the two numbers that multiply together to equal -2016, and add together to equal -20, are -56 and 36, as illustrated here:

-56 × 36 = -2016
-56 + 36 = -20

Now, we can fill in the last two squares in our box with -56x and 36x. Place -56x in the upper left square, and place 36x in the lower right square.

-14	-56x	-42
12x	48x²	36x
	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 -56x and -42. The greatest common factor of -56x and -42 is -14. Therefore, we write -14 to the left of the top row. You can see it here in the color green:

-14	-56x	-42
12x	48x²	36x
	4x	3

Next, let’s look at the bottom row. We have the terms 48x² and 36x. The greatest common factor of 48x² and 36x is 12x. Therefore, we write 12x to the left of the bottom row. You can see it here in the color blue:

-14	-56x	-42
12x	48x²	36x
	4x	3

To find the values below the table, we first divide 48x² by 12x (labeled in blue). This gives us 4x.

48x² ÷ 12x = 4x

You can see this value colored in orange below:

-14	-56x	-42
12x	48x²	36x
	4x	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:

-14	-56x	-42
12x	48x²	36x
	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 48x² - 20x - 42. 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 - 14)(4x + 3)

That’s it! Now you know how to factor the equation 48x² - 20x - 42.

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