Factor 25x² - 70x

Factor 25x² - 70x - 15

Here we will show you how to factor the quadratic function 25x² - 70x - 15 using the box method. In other words, we will show you how to factor 25x squared minus 70x minus 15 (25x^2 - 70x - 15) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 25x² - 70x - 15, like this:

a = 25
b = -70
c = -15

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

-15	-75x	-15
5x	25x²	5x
	5x	1

We put 25x² (a) in the bottom left square and -15 (c) in the top right square, like this:

-15	-75x	-15
5x	25x²	5x
	5x	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 -15 (a × c), and add together to equal -70 (b).

More specifically, 25 times -15 is -375. Therefore, we need to find the two numbers that multiply to equal -375, and add to equal -70.

? × ? = -375
? + ? = -70

After looking at this problem, we can see that the two numbers that multiply together to equal -375, and add together to equal -70, are -75 and 5, as illustrated here:

-75 × 5 = -375
-75 + 5 = -70

Now, we can fill in the last two squares in our box with -75x and 5x. Place -75x in the upper left square, and place 5x in the lower right square.

-15	-75x	-15
5x	25x²	5x
	5x	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 -75x and -15. The greatest common factor of -75x and -15 is -15. Therefore, we write -15 to the left of the top row. You can see it here in the color green:

-15	-75x	-15
5x	25x²	5x
	5x	1

Next, let’s look at the bottom row. We have the terms 25x² and 5x. The greatest common factor of 25x² and 5x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

-15	-75x	-15
5x	25x²	5x
	5x	1

To find the values below the table, we first divide 25x² by 5x (labeled in blue). This gives us 5x.

25x² ÷ 5x = 5x

You can see this value colored in orange below:

-15	-75x	-15
5x	25x²	5x
	5x	1

Next, we divide 5x by 5x (labeled in blue). This gives us 1.

5x ÷ 5x = 1

You can see this value colored in purple below:

-15	-75x	-15
5x	25x²	5x
	5x	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 - 70x - 15. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(5x - 15)(5x + 1)

That’s it! Now you know how to factor the equation 25x² - 70x - 15.

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