Factor -21x² - 59x

Factor -21x² - 59x - 38

Here we will show you how to factor the quadratic function -21x² - 59x - 38 using the box method. In other words, we will show you how to factor negative 21x squared minus 59x minus 38 (-21x^2 - 59x - 38) 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 21x² + 59x + 38. Now we can label the different parts of our equation, like this:

a = 21
b = 59
c = 38

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

1	21x	38
x	21x²	38x
	21x	38

We put 21x² (a) in the bottom left square and 38 (c) in the top right square, like this:

1	21x	38
x	21x²	38x
	21x	38

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 21 times 38 (a × c), and add together to equal 59 (b).

More specifically, 21 times 38 is 798. Therefore, we need to find the two numbers that multiply to equal 798, and add to equal 59.

? × ? = 798
? + ? = 59

After looking at this problem, we can see that the two numbers that multiply together to equal 798, and add together to equal 59, are 21 and 38, as illustrated here:

21 × 38 = 798
21 + 38 = 59

Now, we can fill in the last two squares in our box with 21x and 38x. Place 21x in the upper left square, and place 38x in the lower right square.

1	21x	38
x	21x²	38x
	21x	38

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 21x and 38. The greatest common factor of 21x and 38 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	21x	38
x	21x²	38x
	21x	38

Next, let’s look at the bottom row. We have the terms 21x² and 38x. The greatest common factor of 21x² and 38x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	21x	38
x	21x²	38x
	21x	38

To find the values below the table, we first divide 21x² by x (labeled in blue). This gives us 21x.

21x² ÷ x = 21x

You can see this value colored in orange below:

1	21x	38
x	21x²	38x
	21x	38

Next, we divide 38x by x (labeled in blue). This gives us 38.

38x ÷ x = 38

You can see this value colored in purple below:

1	21x	38
x	21x²	38x
	21x	38

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 -21x² - 59x - 38. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(x + 1)(21x + 38)

In our original quadratic equation, -21x² - 59x - 38, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 1)(21x + 38)

That’s it! Now you know how to factor the equation -21x² - 59x - 38.

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