Factor -25x² + 76x

Factor -25x² + 76x - 51

Here we will show you how to factor the quadratic function -25x² + 76x - 51 using the box method. In other words, we will show you how to factor negative 25x squared plus 76x minus 51 (-25x^2 + 76x - 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 25x² - 76x + 51. Now we can label the different parts of our equation, like this:

a = 25
b = -76
c = 51

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

-51	-51x	51
25x	25x²	-25x
	x	-1

We put 25x² (a) in the bottom left square and 51 (c) in the top right square, like this:

-51	-51x	51
25x	25x²	-25x
	x	-1

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 25 times 51 (a × c), and add together to equal -76 (b).

More specifically, 25 times 51 is 1275. Therefore, we need to find the two numbers that multiply to equal 1275, and add to equal -76.

? × ? = 1275
? + ? = -76

After looking at this problem, we can see that the two numbers that multiply together to equal 1275, and add together to equal -76, are -51 and -25, as illustrated here:

-51 × -25 = 1275
-51 + -25 = -76

Now, we can fill in the last two squares in our box with -51x and -25x. Place -51x in the upper left square, and place -25x in the lower right square.

-51	-51x	51
25x	25x²	-25x
	x	-1

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

-51	-51x	51
25x	25x²	-25x
	x	-1

Next, let’s look at the bottom row. We have the terms 25x² and -25x. The greatest common factor of 25x² and -25x is 25x. Therefore, we write 25x to the left of the bottom row. You can see it here in the color blue:

-51	-51x	51
25x	25x²	-25x
	x	-1

To find the values below the table, we first divide 25x² by 25x (labeled in blue). This gives us x.

25x² ÷ 25x = x

You can see this value colored in orange below:

-51	-51x	51
25x	25x²	-25x
	x	-1

Next, we divide -25x by 25x (labeled in blue). This gives us -1.

-25x ÷ 25x = -1

You can see this value colored in purple below:

-51	-51x	51
25x	25x²	-25x
	x	-1

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 -25x² + 76x - 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:

(25x - 51)(x - 1)

In our original quadratic equation, -25x² + 76x - 51, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(25x - 51)(x - 1)

That’s it! Now you know how to factor the equation -25x² + 76x - 51.

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