Factor -10x² - 41x + 51

Here we will show you how to factor the quadratic function -10x² - 41x + 51 using the box method. In other words, we will show you how to factor negative 10x squared minus 41x plus 51 (-10x^2 - 41x + 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 10x² + 41x - 51. Now we can label the different parts of our equation, like this:

a = 10
b = 41
c = -51

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

-1	-10x	-51
x	10x²	51x
	10x	51

We put 10x² (a) in the bottom left square and -51 (c) in the top right square, like this:

-1	-10x	-51
x	10x²	51x
	10x	51

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 10 times -51 (a × c), and add together to equal 41 (b).

More specifically, 10 times -51 is -510. Therefore, we need to find the two numbers that multiply to equal -510, and add to equal 41.

? × ? = -510
? + ? = 41

After looking at this problem, we can see that the two numbers that multiply together to equal -510, and add together to equal 41, are -10 and 51, as illustrated here:

-10 × 51 = -510
-10 + 51 = 41

Now, we can fill in the last two squares in our box with -10x and 51x. Place -10x in the upper left square, and place 51x in the lower right square.

-1	-10x	-51
x	10x²	51x
	10x	51

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 -51. The greatest common factor of -10x and -51 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-10x	-51
x	10x²	51x
	10x	51

Next, let’s look at the bottom row. We have the terms 10x² and 51x. The greatest common factor of 10x² and 51x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-10x	-51
x	10x²	51x
	10x	51

To find the values below the table, we first divide 10x² by x (labeled in blue). This gives us 10x.

10x² ÷ x = 10x

You can see this value colored in orange below:

-1	-10x	-51
x	10x²	51x
	10x	51

Next, we divide 51x by x (labeled in blue). This gives us 51.

51x ÷ x = 51

You can see this value colored in purple below:

-1	-10x	-51
x	10x²	51x
	10x	51

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 -10x² - 41x + 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:

(x - 1)(10x + 51)

In our original quadratic equation, -10x² - 41x + 51, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 1)(10x + 51)

That’s it! Now you know how to factor the equation -10x² - 41x + 51.

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