Factor 48x² - 20x

Factor 48x² - 20x - 78

Here we will show you how to factor the quadratic function 48x² - 20x - 78 using the box method. In other words, we will show you how to factor 48x squared minus 20x minus 78 (48x^2 - 20x - 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 48x² - 20x - 78, like this:

a = 48
b = -20
c = -78

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

-6	-72x	-78
4x	48x²	52x
	12x	13

We put 48x² (a) in the bottom left square and -78 (c) in the top right square, like this:

-6	-72x	-78
4x	48x²	52x
	12x	13

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 -78 (a × c), and add together to equal -20 (b).

More specifically, 48 times -78 is -3744. Therefore, we need to find the two numbers that multiply to equal -3744, and add to equal -20.

? × ? = -3744
? + ? = -20

After looking at this problem, we can see that the two numbers that multiply together to equal -3744, and add together to equal -20, are -72 and 52, as illustrated here:

-72 × 52 = -3744
-72 + 52 = -20

Now, we can fill in the last two squares in our box with -72x and 52x. Place -72x in the upper left square, and place 52x in the lower right square.

-6	-72x	-78
4x	48x²	52x
	12x	13

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 -72x and -78. The greatest common factor of -72x and -78 is -6. Therefore, we write -6 to the left of the top row. You can see it here in the color green:

-6	-72x	-78
4x	48x²	52x
	12x	13

Next, let’s look at the bottom row. We have the terms 48x² and 52x. The greatest common factor of 48x² and 52x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-6	-72x	-78
4x	48x²	52x
	12x	13

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

48x² ÷ 4x = 12x

You can see this value colored in orange below:

-6	-72x	-78
4x	48x²	52x
	12x	13

Next, we divide 52x by 4x (labeled in blue). This gives us 13.

52x ÷ 4x = 13

You can see this value colored in purple below:

-6	-72x	-78
4x	48x²	52x
	12x	13

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 - 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:

(4x - 6)(12x + 13)

That’s it! Now you know how to factor the equation 48x² - 20x - 78.

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