Factor -36x² - 67x

Factor -36x² - 67x - 31

Here we will show you how to factor the quadratic function -36x² - 67x - 31 using the box method. In other words, we will show you how to factor negative 36x squared minus 67x minus 31 (-36x^2 - 67x - 31) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 36x² + 67x + 31. Now we can label the different parts of our equation, like this:

a = 36
b = 67
c = 31

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

31	31x	31
36x	36x²	36x
	x	1

We put 36x² (a) in the bottom left square and 31 (c) in the top right square, like this:

31	31x	31
36x	36x²	36x
	x	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 36 times 31 (a × c), and add together to equal 67 (b).

More specifically, 36 times 31 is 1116. Therefore, we need to find the two numbers that multiply to equal 1116, and add to equal 67.

? × ? = 1116
? + ? = 67

After looking at this problem, we can see that the two numbers that multiply together to equal 1116, and add together to equal 67, are 31 and 36, as illustrated here:

31 × 36 = 1116
31 + 36 = 67

Now, we can fill in the last two squares in our box with 31x and 36x. Place 31x in the upper left square, and place 36x in the lower right square.

31	31x	31
36x	36x²	36x
	x	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 31x and 31. The greatest common factor of 31x and 31 is 31. Therefore, we write 31 to the left of the top row. You can see it here in the color green:

31	31x	31
36x	36x²	36x
	x	1

Next, let’s look at the bottom row. We have the terms 36x² and 36x. The greatest common factor of 36x² and 36x is 36x. Therefore, we write 36x to the left of the bottom row. You can see it here in the color blue:

31	31x	31
36x	36x²	36x
	x	1

To find the values below the table, we first divide 36x² by 36x (labeled in blue). This gives us x.

36x² ÷ 36x = x

You can see this value colored in orange below:

31	31x	31
36x	36x²	36x
	x	1

Next, we divide 36x by 36x (labeled in blue). This gives us 1.

36x ÷ 36x = 1

You can see this value colored in purple below:

31	31x	31
36x	36x²	36x
	x	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 -36x² - 67x - 31. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(36x + 31)(x + 1)

In our original quadratic equation, -36x² - 67x - 31, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(36x + 31)(x + 1)

That’s it! Now you know how to factor the equation -36x² - 67x - 31.

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Factor -36x² - 67x - 30
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