Factor 56x² - 66x + 10

Here we will show you how to factor the quadratic function 56x² - 66x + 10 using the box method. In other words, we will show you how to factor 56x squared minus 66x plus 10 (56x^2 - 66x + 10) 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 56x² - 66x + 10, like this:

a = 56
b = -66
c = 10

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

-2	-56x	10
2x	56x²	-10x
	28x	-5

We put 56x² (a) in the bottom left square and 10 (c) in the top right square, like this:

-2	-56x	10
2x	56x²	-10x
	28x	-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 56 times 10 (a × c), and add together to equal -66 (b).

More specifically, 56 times 10 is 560. Therefore, we need to find the two numbers that multiply to equal 560, and add to equal -66.

? × ? = 560
? + ? = -66

After looking at this problem, we can see that the two numbers that multiply together to equal 560, and add together to equal -66, are -56 and -10, as illustrated here:

-56 × -10 = 560
-56 + -10 = -66

Now, we can fill in the last two squares in our box with -56x and -10x. Place -56x in the upper left square, and place -10x in the lower right square.

-2	-56x	10
2x	56x²	-10x
	28x	-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 -56x and 10. The greatest common factor of -56x and 10 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-56x	10
2x	56x²	-10x
	28x	-5

Next, let’s look at the bottom row. We have the terms 56x² and -10x. The greatest common factor of 56x² and -10x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-2	-56x	10
2x	56x²	-10x
	28x	-5

To find the values below the table, we first divide 56x² by 2x (labeled in blue). This gives us 28x.

56x² ÷ 2x = 28x

You can see this value colored in orange below:

-2	-56x	10
2x	56x²	-10x
	28x	-5

Next, we divide -10x by 2x (labeled in blue). This gives us -5.

-10x ÷ 2x = -5

You can see this value colored in purple below:

-2	-56x	10
2x	56x²	-10x
	28x	-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 56x² - 66x + 10. 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:

(2x - 2)(28x - 5)

That’s it! Now you know how to factor the equation 56x² - 66x + 10.

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