Factor -8x² - 78x

Factor -8x² - 78x - 85

Here we will show you how to factor the quadratic function -8x² - 78x - 85 using the box method. In other words, we will show you how to factor negative 8x squared minus 78x minus 85 (-8x^2 - 78x - 85) 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 8x² + 78x + 85. Now we can label the different parts of our equation, like this:

a = 8
b = 78
c = 85

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

5	10x	85
4x	8x²	68x
	2x	17

We put 8x² (a) in the bottom left square and 85 (c) in the top right square, like this:

5	10x	85
4x	8x²	68x
	2x	17

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 8 times 85 (a × c), and add together to equal 78 (b).

More specifically, 8 times 85 is 680. Therefore, we need to find the two numbers that multiply to equal 680, and add to equal 78.

? × ? = 680
? + ? = 78

After looking at this problem, we can see that the two numbers that multiply together to equal 680, and add together to equal 78, are 10 and 68, as illustrated here:

10 × 68 = 680
10 + 68 = 78

Now, we can fill in the last two squares in our box with 10x and 68x. Place 10x in the upper left square, and place 68x in the lower right square.

5	10x	85
4x	8x²	68x
	2x	17

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 85. The greatest common factor of 10x and 85 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	10x	85
4x	8x²	68x
	2x	17

Next, let’s look at the bottom row. We have the terms 8x² and 68x. The greatest common factor of 8x² and 68x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

5	10x	85
4x	8x²	68x
	2x	17

To find the values below the table, we first divide 8x² by 4x (labeled in blue). This gives us 2x.

8x² ÷ 4x = 2x

You can see this value colored in orange below:

5	10x	85
4x	8x²	68x
	2x	17

Next, we divide 68x by 4x (labeled in blue). This gives us 17.

68x ÷ 4x = 17

You can see this value colored in purple below:

5	10x	85
4x	8x²	68x
	2x	17

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 -8x² - 78x - 85. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(4x + 5)(2x + 17)

In our original quadratic equation, -8x² - 78x - 85, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 5)(2x + 17)

That’s it! Now you know how to factor the equation -8x² - 78x - 85.

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