Factor -86x² - 85x

Factor -86x² - 85x - 21

Here we will show you how to factor the quadratic function -86x² - 85x - 21 using the box method. In other words, we will show you how to factor negative 86x squared minus 85x minus 21 (-86x^2 - 85x - 21) 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 86x² + 85x + 21. Now we can label the different parts of our equation, like this:

a = 86
b = 85
c = 21

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

21	42x	21
43x	86x²	43x
	2x	1

We put 86x² (a) in the bottom left square and 21 (c) in the top right square, like this:

21	42x	21
43x	86x²	43x
	2x	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 86 times 21 (a × c), and add together to equal 85 (b).

More specifically, 86 times 21 is 1806. Therefore, we need to find the two numbers that multiply to equal 1806, and add to equal 85.

? × ? = 1806
? + ? = 85

After looking at this problem, we can see that the two numbers that multiply together to equal 1806, and add together to equal 85, are 42 and 43, as illustrated here:

42 × 43 = 1806
42 + 43 = 85

Now, we can fill in the last two squares in our box with 42x and 43x. Place 42x in the upper left square, and place 43x in the lower right square.

21	42x	21
43x	86x²	43x
	2x	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 42x and 21. The greatest common factor of 42x and 21 is 21. Therefore, we write 21 to the left of the top row. You can see it here in the color green:

21	42x	21
43x	86x²	43x
	2x	1

Next, let’s look at the bottom row. We have the terms 86x² and 43x. The greatest common factor of 86x² and 43x is 43x. Therefore, we write 43x to the left of the bottom row. You can see it here in the color blue:

21	42x	21
43x	86x²	43x
	2x	1

To find the values below the table, we first divide 86x² by 43x (labeled in blue). This gives us 2x.

86x² ÷ 43x = 2x

You can see this value colored in orange below:

21	42x	21
43x	86x²	43x
	2x	1

Next, we divide 43x by 43x (labeled in blue). This gives us 1.

43x ÷ 43x = 1

You can see this value colored in purple below:

21	42x	21
43x	86x²	43x
	2x	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 -86x² - 85x - 21. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(43x + 21)(2x + 1)

In our original quadratic equation, -86x² - 85x - 21, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(43x + 21)(2x + 1)

That’s it! Now you know how to factor the equation -86x² - 85x - 21.

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