Factor -36x² - 75x

Factor -36x² - 75x - 34

Here we will show you how to factor the quadratic function -36x² - 75x - 34 using the box method. In other words, we will show you how to factor negative 36x squared minus 75x minus 34 (-36x^2 - 75x - 34) 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 + 75x + 34. Now we can label the different parts of our equation, like this:

a = 36
b = 75
c = 34

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

2	24x	34
3x	36x²	51x
	12x	17

We put 36x² (a) in the bottom left square and 34 (c) in the top right square, like this:

2	24x	34
3x	36x²	51x
	12x	17

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 34 (a × c), and add together to equal 75 (b).

More specifically, 36 times 34 is 1224. Therefore, we need to find the two numbers that multiply to equal 1224, and add to equal 75.

? × ? = 1224
? + ? = 75

After looking at this problem, we can see that the two numbers that multiply together to equal 1224, and add together to equal 75, are 24 and 51, as illustrated here:

24 × 51 = 1224
24 + 51 = 75

Now, we can fill in the last two squares in our box with 24x and 51x. Place 24x in the upper left square, and place 51x in the lower right square.

2	24x	34
3x	36x²	51x
	12x	17

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 24x and 34. The greatest common factor of 24x and 34 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	24x	34
3x	36x²	51x
	12x	17

Next, let’s look at the bottom row. We have the terms 36x² and 51x. The greatest common factor of 36x² and 51x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

2	24x	34
3x	36x²	51x
	12x	17

To find the values below the table, we first divide 36x² by 3x (labeled in blue). This gives us 12x.

36x² ÷ 3x = 12x

You can see this value colored in orange below:

2	24x	34
3x	36x²	51x
	12x	17

Next, we divide 51x by 3x (labeled in blue). This gives us 17.

51x ÷ 3x = 17

You can see this value colored in purple below:

2	24x	34
3x	36x²	51x
	12x	17

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 - 75x - 34. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(3x + 2)(12x + 17)

In our original quadratic equation, -36x² - 75x - 34, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x + 2)(12x + 17)

That’s it! Now you know how to factor the equation -36x² - 75x - 34.

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