Factor -4x² - 28x + 51

Here we will show you how to factor the quadratic function -4x² - 28x + 51 using the box method. In other words, we will show you how to factor negative 4x squared minus 28x plus 51 (-4x^2 - 28x + 51) 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 4x² + 28x - 51. Now we can label the different parts of our equation, like this:

a = 4
b = 28
c = -51

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

-3	-6x	-51
2x	4x²	34x
	2x	17

We put 4x² (a) in the bottom left square and -51 (c) in the top right square, like this:

-3	-6x	-51
2x	4x²	34x
	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 4 times -51 (a × c), and add together to equal 28 (b).

More specifically, 4 times -51 is -204. Therefore, we need to find the two numbers that multiply to equal -204, and add to equal 28.

? × ? = -204
? + ? = 28

After looking at this problem, we can see that the two numbers that multiply together to equal -204, and add together to equal 28, are -6 and 34, as illustrated here:

-6 × 34 = -204
-6 + 34 = 28

Now, we can fill in the last two squares in our box with -6x and 34x. Place -6x in the upper left square, and place 34x in the lower right square.

-3	-6x	-51
2x	4x²	34x
	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 -6x and -51. The greatest common factor of -6x and -51 is -3. Therefore, we write -3 to the left of the top row. You can see it here in the color green:

-3	-6x	-51
2x	4x²	34x
	2x	17

Next, let’s look at the bottom row. We have the terms 4x² and 34x. The greatest common factor of 4x² and 34x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-3	-6x	-51
2x	4x²	34x
	2x	17

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

4x² ÷ 2x = 2x

You can see this value colored in orange below:

-3	-6x	-51
2x	4x²	34x
	2x	17

Next, we divide 34x by 2x (labeled in blue). This gives us 17.

34x ÷ 2x = 17

You can see this value colored in purple below:

-3	-6x	-51
2x	4x²	34x
	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 -4x² - 28x + 51. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(2x - 3)(2x + 17)

In our original quadratic equation, -4x² - 28x + 51, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x - 3)(2x + 17)

That’s it! Now you know how to factor the equation -4x² - 28x + 51.

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