Factor -100x² - 89x

Factor -100x² - 89x - 16

Here we will show you how to factor the quadratic function -100x² - 89x - 16 using the box method. In other words, we will show you how to factor negative 100x squared minus 89x minus 16 (-100x^2 - 89x - 16) 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 100x² + 89x + 16. Now we can label the different parts of our equation, like this:

a = 100
b = 89
c = 16

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

1	25x	16
4x	100x²	64x
	25x	16

We put 100x² (a) in the bottom left square and 16 (c) in the top right square, like this:

1	25x	16
4x	100x²	64x
	25x	16

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 100 times 16 (a × c), and add together to equal 89 (b).

More specifically, 100 times 16 is 1600. Therefore, we need to find the two numbers that multiply to equal 1600, and add to equal 89.

? × ? = 1600
? + ? = 89

After looking at this problem, we can see that the two numbers that multiply together to equal 1600, and add together to equal 89, are 25 and 64, as illustrated here:

25 × 64 = 1600
25 + 64 = 89

Now, we can fill in the last two squares in our box with 25x and 64x. Place 25x in the upper left square, and place 64x in the lower right square.

1	25x	16
4x	100x²	64x
	25x	16

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 25x and 16. The greatest common factor of 25x and 16 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	25x	16
4x	100x²	64x
	25x	16

Next, let’s look at the bottom row. We have the terms 100x² and 64x. The greatest common factor of 100x² and 64x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

1	25x	16
4x	100x²	64x
	25x	16

To find the values below the table, we first divide 100x² by 4x (labeled in blue). This gives us 25x.

100x² ÷ 4x = 25x

You can see this value colored in orange below:

1	25x	16
4x	100x²	64x
	25x	16

Next, we divide 64x by 4x (labeled in blue). This gives us 16.

64x ÷ 4x = 16

You can see this value colored in purple below:

1	25x	16
4x	100x²	64x
	25x	16

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 -100x² - 89x - 16. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(4x + 1)(25x + 16)

In our original quadratic equation, -100x² - 89x - 16, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 1)(25x + 16)

That’s it! Now you know how to factor the equation -100x² - 89x - 16.

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