Factor -25x² + 68x + 93

Here we will show you how to factor the quadratic function -25x² + 68x + 93 using the box method. In other words, we will show you how to factor negative 25x squared plus 68x plus 93 (-25x^2 + 68x + 93) 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 - 68x - 93. Now we can label the different parts of our equation, like this:

a = 25
b = -68
c = -93

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

-93	-93x	-93
25x	25x²	25x
	x	1

We put 25x² (a) in the bottom left square and -93 (c) in the top right square, like this:

-93	-93x	-93
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 -93 (a × c), and add together to equal -68 (b).

More specifically, 25 times -93 is -2325. Therefore, we need to find the two numbers that multiply to equal -2325, and add to equal -68.

? × ? = -2325
? + ? = -68

After looking at this problem, we can see that the two numbers that multiply together to equal -2325, and add together to equal -68, are -93 and 25, as illustrated here:

-93 × 25 = -2325
-93 + 25 = -68

Now, we can fill in the last two squares in our box with -93x and 25x. Place -93x in the upper left square, and place 25x in the lower right square.

-93	-93x	-93
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 -93x and -93. The greatest common factor of -93x and -93 is -93. Therefore, we write -93 to the left of the top row. You can see it here in the color green:

-93	-93x	-93
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:

-93	-93x	-93
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:

-93	-93x	-93
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:

-93	-93x	-93
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 + 68x + 93. 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 - 93)(x + 1)

In our original quadratic equation, -25x² + 68x + 93, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(25x - 93)(x + 1)

That’s it! Now you know how to factor the equation -25x² + 68x + 93.

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