Factor -86x² - 97x

Factor -86x² - 97x - 27

Here we will show you how to factor the quadratic function -86x² - 97x - 27 using the box method. In other words, we will show you how to factor negative 86x squared minus 97x minus 27 (-86x^2 - 97x - 27) 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 + 97x + 27. Now we can label the different parts of our equation, like this:

a = 86
b = 97
c = 27

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

1	43x	27
2x	86x²	54x
	43x	27

We put 86x² (a) in the bottom left square and 27 (c) in the top right square, like this:

1	43x	27
2x	86x²	54x
	43x	27

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 27 (a × c), and add together to equal 97 (b).

More specifically, 86 times 27 is 2322. Therefore, we need to find the two numbers that multiply to equal 2322, and add to equal 97.

? × ? = 2322
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 2322, and add together to equal 97, are 43 and 54, as illustrated here:

43 × 54 = 2322
43 + 54 = 97

Now, we can fill in the last two squares in our box with 43x and 54x. Place 43x in the upper left square, and place 54x in the lower right square.

1	43x	27
2x	86x²	54x
	43x	27

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 43x and 27. The greatest common factor of 43x and 27 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	43x	27
2x	86x²	54x
	43x	27

Next, let’s look at the bottom row. We have the terms 86x² and 54x. The greatest common factor of 86x² and 54x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	43x	27
2x	86x²	54x
	43x	27

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

86x² ÷ 2x = 43x

You can see this value colored in orange below:

1	43x	27
2x	86x²	54x
	43x	27

Next, we divide 54x by 2x (labeled in blue). This gives us 27.

54x ÷ 2x = 27

You can see this value colored in purple below:

1	43x	27
2x	86x²	54x
	43x	27

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 - 97x - 27. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(2x + 1)(43x + 27)

In our original quadratic equation, -86x² - 97x - 27, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x + 1)(43x + 27)

That’s it! Now you know how to factor the equation -86x² - 97x - 27.

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