Factor 24x² - 70x + 11

Here we will show you how to factor the quadratic function 24x² - 70x + 11 using the box method. In other words, we will show you how to factor 24x squared minus 70x plus 11 (24x^2 - 70x + 11) 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 24x² - 70x + 11, like this:

a = 24
b = -70
c = 11

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

-11	-66x	11
4x	24x²	-4x
	6x	-1

We put 24x² (a) in the bottom left square and 11 (c) in the top right square, like this:

-11	-66x	11
4x	24x²	-4x
	6x	-1

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 24 times 11 (a × c), and add together to equal -70 (b).

More specifically, 24 times 11 is 264. Therefore, we need to find the two numbers that multiply to equal 264, and add to equal -70.

? × ? = 264
? + ? = -70

After looking at this problem, we can see that the two numbers that multiply together to equal 264, and add together to equal -70, are -66 and -4, as illustrated here:

-66 × -4 = 264
-66 + -4 = -70

Now, we can fill in the last two squares in our box with -66x and -4x. Place -66x in the upper left square, and place -4x in the lower right square.

-11	-66x	11
4x	24x²	-4x
	6x	-1

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 -66x and 11. The greatest common factor of -66x and 11 is -11. Therefore, we write -11 to the left of the top row. You can see it here in the color green:

-11	-66x	11
4x	24x²	-4x
	6x	-1

Next, let’s look at the bottom row. We have the terms 24x² and -4x. The greatest common factor of 24x² and -4x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-11	-66x	11
4x	24x²	-4x
	6x	-1

To find the values below the table, we first divide 24x² by 4x (labeled in blue). This gives us 6x.

24x² ÷ 4x = 6x

You can see this value colored in orange below:

-11	-66x	11
4x	24x²	-4x
	6x	-1

Next, we divide -4x by 4x (labeled in blue). This gives us -1.

-4x ÷ 4x = -1

You can see this value colored in purple below:

-11	-66x	11
4x	24x²	-4x
	6x	-1

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 24x² - 70x + 11. 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 - 11)(6x - 1)

That’s it! Now you know how to factor the equation 24x² - 70x + 11.

Factoring Quadratics
Go here if you want to learn how to factor another quadratic function.

Factor 24x² - 70x + 16
Here is the next quadratic function on our list that we have factored for you.