Factor -4x² - 27x

Factor -4x² - 27x - 45

Here we will show you how to factor the quadratic function -4x² - 27x - 45 using the box method. In other words, we will show you how to factor negative 4x squared minus 27x minus 45 (-4x^2 - 27x - 45) 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 4x² + 27x + 45. Now we can label the different parts of our equation, like this:

a = 4
b = 27
c = 45

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

3	12x	45
x	4x²	15x
	4x	15

We put 4x² (a) in the bottom left square and 45 (c) in the top right square, like this:

3	12x	45
x	4x²	15x
	4x	15

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 4 times 45 (a × c), and add together to equal 27 (b).

More specifically, 4 times 45 is 180. Therefore, we need to find the two numbers that multiply to equal 180, and add to equal 27.

? × ? = 180
? + ? = 27

After looking at this problem, we can see that the two numbers that multiply together to equal 180, and add together to equal 27, are 12 and 15, as illustrated here:

12 × 15 = 180
12 + 15 = 27

Now, we can fill in the last two squares in our box with 12x and 15x. Place 12x in the upper left square, and place 15x in the lower right square.

3	12x	45
x	4x²	15x
	4x	15

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 12x and 45. The greatest common factor of 12x and 45 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	12x	45
x	4x²	15x
	4x	15

Next, let’s look at the bottom row. We have the terms 4x² and 15x. The greatest common factor of 4x² and 15x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

3	12x	45
x	4x²	15x
	4x	15

To find the values below the table, we first divide 4x² by x (labeled in blue). This gives us 4x.

4x² ÷ x = 4x

You can see this value colored in orange below:

3	12x	45
x	4x²	15x
	4x	15

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

15x ÷ x = 15

You can see this value colored in purple below:

3	12x	45
x	4x²	15x
	4x	15

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 -4x² - 27x - 45. 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 + 3)(4x + 15)

In our original quadratic equation, -4x² - 27x - 45, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 3)(4x + 15)

That’s it! Now you know how to factor the equation -4x² - 27x - 45.

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Factor -4x² - 27x - 44
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