Factor -7x² - 37x

Factor -7x² - 37x - 30

Here we will show you how to factor the quadratic function -7x² - 37x - 30 using the box method. In other words, we will show you how to factor negative 7x squared minus 37x minus 30 (-7x^2 - 37x - 30) 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 7x² + 37x + 30. Now we can label the different parts of our equation, like this:

a = 7
b = 37
c = 30

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

1	7x	30
x	7x²	30x
	7x	30

We put 7x² (a) in the bottom left square and 30 (c) in the top right square, like this:

1	7x	30
x	7x²	30x
	7x	30

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 7 times 30 (a × c), and add together to equal 37 (b).

More specifically, 7 times 30 is 210. Therefore, we need to find the two numbers that multiply to equal 210, and add to equal 37.

? × ? = 210
? + ? = 37

After looking at this problem, we can see that the two numbers that multiply together to equal 210, and add together to equal 37, are 7 and 30, as illustrated here:

7 × 30 = 210
7 + 30 = 37

Now, we can fill in the last two squares in our box with 7x and 30x. Place 7x in the upper left square, and place 30x in the lower right square.

1	7x	30
x	7x²	30x
	7x	30

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 7x and 30. The greatest common factor of 7x and 30 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	7x	30
x	7x²	30x
	7x	30

Next, let’s look at the bottom row. We have the terms 7x² and 30x. The greatest common factor of 7x² and 30x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	7x	30
x	7x²	30x
	7x	30

To find the values below the table, we first divide 7x² by x (labeled in blue). This gives us 7x.

7x² ÷ x = 7x

You can see this value colored in orange below:

1	7x	30
x	7x²	30x
	7x	30

Next, we divide 30x by x (labeled in blue). This gives us 30.

30x ÷ x = 30

You can see this value colored in purple below:

1	7x	30
x	7x²	30x
	7x	30

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 -7x² - 37x - 30. 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)(7x + 30)

In our original quadratic equation, -7x² - 37x - 30, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 1)(7x + 30)

That’s it! Now you know how to factor the equation -7x² - 37x - 30.

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