Factor -5x² - 22x + 64

Here we will show you how to factor the quadratic function -5x² - 22x + 64 using the box method. In other words, we will show you how to factor negative 5x squared minus 22x plus 64 (-5x^2 - 22x + 64) 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 5x² + 22x - 64. Now we can label the different parts of our equation, like this:

a = 5
b = 22
c = -64

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

-2	-10x	-64
x	5x²	32x
	5x	32

We put 5x² (a) in the bottom left square and -64 (c) in the top right square, like this:

-2	-10x	-64
x	5x²	32x
	5x	32

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 5 times -64 (a × c), and add together to equal 22 (b).

More specifically, 5 times -64 is -320. Therefore, we need to find the two numbers that multiply to equal -320, and add to equal 22.

? × ? = -320
? + ? = 22

After looking at this problem, we can see that the two numbers that multiply together to equal -320, and add together to equal 22, are -10 and 32, as illustrated here:

-10 × 32 = -320
-10 + 32 = 22

Now, we can fill in the last two squares in our box with -10x and 32x. Place -10x in the upper left square, and place 32x in the lower right square.

-2	-10x	-64
x	5x²	32x
	5x	32

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 -10x and -64. The greatest common factor of -10x and -64 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-10x	-64
x	5x²	32x
	5x	32

Next, let’s look at the bottom row. We have the terms 5x² and 32x. The greatest common factor of 5x² and 32x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-2	-10x	-64
x	5x²	32x
	5x	32

To find the values below the table, we first divide 5x² by x (labeled in blue). This gives us 5x.

5x² ÷ x = 5x

You can see this value colored in orange below:

-2	-10x	-64
x	5x²	32x
	5x	32

Next, we divide 32x by x (labeled in blue). This gives us 32.

32x ÷ x = 32

You can see this value colored in purple below:

-2	-10x	-64
x	5x²	32x
	5x	32

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 -5x² - 22x + 64. 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)(5x + 32)

In our original quadratic equation, -5x² - 22x + 64, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 2)(5x + 32)

That’s it! Now you know how to factor the equation -5x² - 22x + 64.

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