Factor -3x² - 14x + 40

Here we will show you how to factor the quadratic function -3x² - 14x + 40 using the box method. In other words, we will show you how to factor negative 3x squared minus 14x plus 40 (-3x^2 - 14x + 40) 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 3x² + 14x - 40. Now we can label the different parts of our equation, like this:

a = 3
b = 14
c = -40

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

-2	-6x	-40
x	3x²	20x
	3x	20

We put 3x² (a) in the bottom left square and -40 (c) in the top right square, like this:

-2	-6x	-40
x	3x²	20x
	3x	20

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 3 times -40 (a × c), and add together to equal 14 (b).

More specifically, 3 times -40 is -120. Therefore, we need to find the two numbers that multiply to equal -120, and add to equal 14.

? × ? = -120
? + ? = 14

After looking at this problem, we can see that the two numbers that multiply together to equal -120, and add together to equal 14, are -6 and 20, as illustrated here:

-6 × 20 = -120
-6 + 20 = 14

Now, we can fill in the last two squares in our box with -6x and 20x. Place -6x in the upper left square, and place 20x in the lower right square.

-2	-6x	-40
x	3x²	20x
	3x	20

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 -6x and -40. The greatest common factor of -6x and -40 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-6x	-40
x	3x²	20x
	3x	20

Next, let’s look at the bottom row. We have the terms 3x² and 20x. The greatest common factor of 3x² and 20x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-2	-6x	-40
x	3x²	20x
	3x	20

To find the values below the table, we first divide 3x² by x (labeled in blue). This gives us 3x.

3x² ÷ x = 3x

You can see this value colored in orange below:

-2	-6x	-40
x	3x²	20x
	3x	20

Next, we divide 20x by x (labeled in blue). This gives us 20.

20x ÷ x = 20

You can see this value colored in purple below:

-2	-6x	-40
x	3x²	20x
	3x	20

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 -3x² - 14x + 40. 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 - 2)(3x + 20)

In our original quadratic equation, -3x² - 14x + 40, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 2)(3x + 20)

That’s it! Now you know how to factor the equation -3x² - 14x + 40.

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