Factor -15x² - 54x

Factor -15x² - 54x - 27

Here we will show you how to factor the quadratic function -15x² - 54x - 27 using the box method. In other words, we will show you how to factor negative 15x squared minus 54x minus 27 (-15x^2 - 54x - 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 15x² + 54x + 27. Now we can label the different parts of our equation, like this:

a = 15
b = 54
c = 27

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

9	9x	27
15x	15x²	45x
	x	3

We put 15x² (a) in the bottom left square and 27 (c) in the top right square, like this:

9	9x	27
15x	15x²	45x
	x	3

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 15 times 27 (a × c), and add together to equal 54 (b).

More specifically, 15 times 27 is 405. Therefore, we need to find the two numbers that multiply to equal 405, and add to equal 54.

? × ? = 405
? + ? = 54

After looking at this problem, we can see that the two numbers that multiply together to equal 405, and add together to equal 54, are 9 and 45, as illustrated here:

9 × 45 = 405
9 + 45 = 54

Now, we can fill in the last two squares in our box with 9x and 45x. Place 9x in the upper left square, and place 45x in the lower right square.

9	9x	27
15x	15x²	45x
	x	3

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 9x and 27. The greatest common factor of 9x and 27 is 9. Therefore, we write 9 to the left of the top row. You can see it here in the color green:

9	9x	27
15x	15x²	45x
	x	3

Next, let’s look at the bottom row. We have the terms 15x² and 45x. The greatest common factor of 15x² and 45x is 15x. Therefore, we write 15x to the left of the bottom row. You can see it here in the color blue:

9	9x	27
15x	15x²	45x
	x	3

To find the values below the table, we first divide 15x² by 15x (labeled in blue). This gives us x.

15x² ÷ 15x = x

You can see this value colored in orange below:

9	9x	27
15x	15x²	45x
	x	3

Next, we divide 45x by 15x (labeled in blue). This gives us 3.

45x ÷ 15x = 3

You can see this value colored in purple below:

9	9x	27
15x	15x²	45x
	x	3

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 -15x² - 54x - 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:

(15x + 9)(x + 3)

In our original quadratic equation, -15x² - 54x - 27, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(15x + 9)(x + 3)

That’s it! Now you know how to factor the equation -15x² - 54x - 27.

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