Factor 52x² + 15x

Factor 52x² + 15x - 67

Here we will show you how to factor the quadratic function 52x² + 15x - 67 using the box method. In other words, we will show you how to factor 52x squared plus 15x minus 67 (52x^2 + 15x - 67) 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 52x² + 15x - 67, like this:

a = 52
b = 15
c = -67

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

-1	-52x	-67
x	52x²	67x
	52x	67

We put 52x² (a) in the bottom left square and -67 (c) in the top right square, like this:

-1	-52x	-67
x	52x²	67x
	52x	67

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 52 times -67 (a × c), and add together to equal 15 (b).

More specifically, 52 times -67 is -3484. Therefore, we need to find the two numbers that multiply to equal -3484, and add to equal 15.

? × ? = -3484
? + ? = 15

After looking at this problem, we can see that the two numbers that multiply together to equal -3484, and add together to equal 15, are -52 and 67, as illustrated here:

-52 × 67 = -3484
-52 + 67 = 15

Now, we can fill in the last two squares in our box with -52x and 67x. Place -52x in the upper left square, and place 67x in the lower right square.

-1	-52x	-67
x	52x²	67x
	52x	67

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 -52x and -67. The greatest common factor of -52x and -67 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-52x	-67
x	52x²	67x
	52x	67

Next, let’s look at the bottom row. We have the terms 52x² and 67x. The greatest common factor of 52x² and 67x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-52x	-67
x	52x²	67x
	52x	67

To find the values below the table, we first divide 52x² by x (labeled in blue). This gives us 52x.

52x² ÷ x = 52x

You can see this value colored in orange below:

-1	-52x	-67
x	52x²	67x
	52x	67

Next, we divide 67x by x (labeled in blue). This gives us 67.

67x ÷ x = 67

You can see this value colored in purple below:

-1	-52x	-67
x	52x²	67x
	52x	67

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 52x² + 15x - 67. 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:

(x - 1)(52x + 67)

That’s it! Now you know how to factor the equation 52x² + 15x - 67.

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