Factor -25x² - 35x + 44

Here we will show you how to factor the quadratic function -25x² - 35x + 44 using the box method. In other words, we will show you how to factor negative 25x squared minus 35x plus 44 (-25x^2 - 35x + 44) 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 25x² + 35x - 44. Now we can label the different parts of our equation, like this:

a = 25
b = 35
c = -44

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

-4	-20x	-44
5x	25x²	55x
	5x	11

We put 25x² (a) in the bottom left square and -44 (c) in the top right square, like this:

-4	-20x	-44
5x	25x²	55x
	5x	11

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 25 times -44 (a × c), and add together to equal 35 (b).

More specifically, 25 times -44 is -1100. Therefore, we need to find the two numbers that multiply to equal -1100, and add to equal 35.

? × ? = -1100
? + ? = 35

After looking at this problem, we can see that the two numbers that multiply together to equal -1100, and add together to equal 35, are -20 and 55, as illustrated here:

-20 × 55 = -1100
-20 + 55 = 35

Now, we can fill in the last two squares in our box with -20x and 55x. Place -20x in the upper left square, and place 55x in the lower right square.

-4	-20x	-44
5x	25x²	55x
	5x	11

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 -20x and -44. The greatest common factor of -20x and -44 is -4. Therefore, we write -4 to the left of the top row. You can see it here in the color green:

-4	-20x	-44
5x	25x²	55x
	5x	11

Next, let’s look at the bottom row. We have the terms 25x² and 55x. The greatest common factor of 25x² and 55x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

-4	-20x	-44
5x	25x²	55x
	5x	11

To find the values below the table, we first divide 25x² by 5x (labeled in blue). This gives us 5x.

25x² ÷ 5x = 5x

You can see this value colored in orange below:

-4	-20x	-44
5x	25x²	55x
	5x	11

Next, we divide 55x by 5x (labeled in blue). This gives us 11.

55x ÷ 5x = 11

You can see this value colored in purple below:

-4	-20x	-44
5x	25x²	55x
	5x	11

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 -25x² - 35x + 44. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(5x - 4)(5x + 11)

In our original quadratic equation, -25x² - 35x + 44, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(5x - 4)(5x + 11)

That’s it! Now you know how to factor the equation -25x² - 35x + 44.

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