Factor 60x² + 55x + 10

Here we will show you how to factor the quadratic function 60x² + 55x + 10 using the box method. In other words, we will show you how to factor 60x squared plus 55x plus 10 (60x^2 + 55x + 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 60x² + 55x + 10, like this:

a = 60
b = 55
c = 10

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

5	15x	10
20x	60x²	40x
	3x	2

We put 60x² (a) in the bottom left square and 10 (c) in the top right square, like this:

5	15x	10
20x	60x²	40x
	3x	2

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 60 times 10 (a × c), and add together to equal 55 (b).

More specifically, 60 times 10 is 600. Therefore, we need to find the two numbers that multiply to equal 600, and add to equal 55.

? × ? = 600
? + ? = 55

After looking at this problem, we can see that the two numbers that multiply together to equal 600, and add together to equal 55, are 15 and 40, as illustrated here:

15 × 40 = 600
15 + 40 = 55

Now, we can fill in the last two squares in our box with 15x and 40x. Place 15x in the upper left square, and place 40x in the lower right square.

5	15x	10
20x	60x²	40x
	3x	2

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 15x and 10. The greatest common factor of 15x and 10 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	15x	10
20x	60x²	40x
	3x	2

Next, let’s look at the bottom row. We have the terms 60x² and 40x. The greatest common factor of 60x² and 40x is 20x. Therefore, we write 20x to the left of the bottom row. You can see it here in the color blue:

5	15x	10
20x	60x²	40x
	3x	2

To find the values below the table, we first divide 60x² by 20x (labeled in blue). This gives us 3x.

60x² ÷ 20x = 3x

You can see this value colored in orange below:

5	15x	10
20x	60x²	40x
	3x	2

Next, we divide 40x by 20x (labeled in blue). This gives us 2.

40x ÷ 20x = 2

You can see this value colored in purple below:

5	15x	10
20x	60x²	40x
	3x	2

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 60x² + 55x + 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:

(20x + 5)(3x + 2)

That’s it! Now you know how to factor the equation 60x² + 55x + 10.

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