Factor -96x² - 53x + 78

Here we will show you how to factor the quadratic function -96x² - 53x + 78 using the box method. In other words, we will show you how to factor negative 96x squared minus 53x plus 78 (-96x^2 - 53x + 78) 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 96x² + 53x - 78. Now we can label the different parts of our equation, like this:

a = 96
b = 53
c = -78

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

-2	-64x	-78
3x	96x²	117x
	32x	39

We put 96x² (a) in the bottom left square and -78 (c) in the top right square, like this:

-2	-64x	-78
3x	96x²	117x
	32x	39

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 96 times -78 (a × c), and add together to equal 53 (b).

More specifically, 96 times -78 is -7488. Therefore, we need to find the two numbers that multiply to equal -7488, and add to equal 53.

? × ? = -7488
? + ? = 53

After looking at this problem, we can see that the two numbers that multiply together to equal -7488, and add together to equal 53, are -64 and 117, as illustrated here:

-64 × 117 = -7488
-64 + 117 = 53

Now, we can fill in the last two squares in our box with -64x and 117x. Place -64x in the upper left square, and place 117x in the lower right square.

-2	-64x	-78
3x	96x²	117x
	32x	39

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 -64x and -78. The greatest common factor of -64x and -78 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-64x	-78
3x	96x²	117x
	32x	39

Next, let’s look at the bottom row. We have the terms 96x² and 117x. The greatest common factor of 96x² and 117x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

-2	-64x	-78
3x	96x²	117x
	32x	39

To find the values below the table, we first divide 96x² by 3x (labeled in blue). This gives us 32x.

96x² ÷ 3x = 32x

You can see this value colored in orange below:

-2	-64x	-78
3x	96x²	117x
	32x	39

Next, we divide 117x by 3x (labeled in blue). This gives us 39.

117x ÷ 3x = 39

You can see this value colored in purple below:

-2	-64x	-78
3x	96x²	117x
	32x	39

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 -96x² - 53x + 78. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(3x - 2)(32x + 39)

In our original quadratic equation, -96x² - 53x + 78, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x - 2)(32x + 39)

That’s it! Now you know how to factor the equation -96x² - 53x + 78.

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