Factor -6x² + 37x

Factor -6x² + 37x - 56

Here we will show you how to factor the quadratic function -6x² + 37x - 56 using the box method. In other words, we will show you how to factor negative 6x squared plus 37x minus 56 (-6x^2 + 37x - 56) 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 6x² - 37x + 56. Now we can label the different parts of our equation, like this:

a = 6
b = -37
c = 56

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

-7	-21x	56
2x	6x²	-16x
	3x	-8

We put 6x² (a) in the bottom left square and 56 (c) in the top right square, like this:

-7	-21x	56
2x	6x²	-16x
	3x	-8

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 6 times 56 (a × c), and add together to equal -37 (b).

More specifically, 6 times 56 is 336. Therefore, we need to find the two numbers that multiply to equal 336, and add to equal -37.

? × ? = 336
? + ? = -37

After looking at this problem, we can see that the two numbers that multiply together to equal 336, and add together to equal -37, are -21 and -16, as illustrated here:

-21 × -16 = 336
-21 + -16 = -37

Now, we can fill in the last two squares in our box with -21x and -16x. Place -21x in the upper left square, and place -16x in the lower right square.

-7	-21x	56
2x	6x²	-16x
	3x	-8

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 -21x and 56. The greatest common factor of -21x and 56 is -7. Therefore, we write -7 to the left of the top row. You can see it here in the color green:

-7	-21x	56
2x	6x²	-16x
	3x	-8

Next, let’s look at the bottom row. We have the terms 6x² and -16x. The greatest common factor of 6x² and -16x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-7	-21x	56
2x	6x²	-16x
	3x	-8

To find the values below the table, we first divide 6x² by 2x (labeled in blue). This gives us 3x.

6x² ÷ 2x = 3x

You can see this value colored in orange below:

-7	-21x	56
2x	6x²	-16x
	3x	-8

Next, we divide -16x by 2x (labeled in blue). This gives us -8.

-16x ÷ 2x = -8

You can see this value colored in purple below:

-7	-21x	56
2x	6x²	-16x
	3x	-8

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 -6x² + 37x - 56. 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 - 7)(3x - 8)

In our original quadratic equation, -6x² + 37x - 56, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x - 7)(3x - 8)

That’s it! Now you know how to factor the equation -6x² + 37x - 56.

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Factor -6x² + 37x - 55
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