Factor -x² - x + 72

Here we will show you how to factor the quadratic function -x² - x + 72 using the box method. In other words, we will show you how to factor negative x squared minus x plus 72 (-x^2 - x + 72) 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 x² + x - 72. Now we can label the different parts of our equation, like this:

a = 1
b = 1
c = -72

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

-8	-8x	-72
x	x²	9x
	x	9

We put x² (a) in the bottom left square and -72 (c) in the top right square, like this:

-8	-8x	-72
x	x²	9x
	x	9

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 1 times -72 (a × c), and add together to equal 1 (b).

More specifically, 1 times -72 is -72. Therefore, we need to find the two numbers that multiply to equal -72, and add to equal 1.

? × ? = -72
? + ? = 1

After looking at this problem, we can see that the two numbers that multiply together to equal -72, and add together to equal 1, are -8 and 9, as illustrated here:

-8 × 9 = -72
-8 + 9 = 1

Now, we can fill in the last two squares in our box with -8x and 9x. Place -8x in the upper left square, and place 9x in the lower right square.

-8	-8x	-72
x	x²	9x
	x	9

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 -8x and -72. The greatest common factor of -8x and -72 is -8. Therefore, we write -8 to the left of the top row. You can see it here in the color green:

-8	-8x	-72
x	x²	9x
	x	9

Next, let’s look at the bottom row. We have the terms x² and 9x. The greatest common factor of x² and 9x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-8	-8x	-72
x	x²	9x
	x	9

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

x² ÷ x = x

You can see this value colored in orange below:

-8	-8x	-72
x	x²	9x
	x	9

Next, we divide 9x by x (labeled in blue). This gives us 9.

9x ÷ x = 9

You can see this value colored in purple below:

-8	-8x	-72
x	x²	9x
	x	9

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 -x² - x + 72. 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 - 8)(x + 9)

In our original quadratic equation, -x² - x + 72, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 8)(x + 9)

That’s it! Now you know how to factor the equation -x² - x + 72.

Factoring Quadratics
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Factor -x² - 1x + 90
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