Factor 7x² - 3x

Factor 7x² - 3x - 100

Here we will show you how to factor the quadratic function 7x² - 3x - 100 using the box method. In other words, we will show you how to factor 7x squared minus 3x minus 100 (7x^2 - 3x - 100) 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 7x² - 3x - 100, like this:

a = 7
b = -3
c = -100

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

-4	-28x	-100
x	7x²	25x
	7x	25

We put 7x² (a) in the bottom left square and -100 (c) in the top right square, like this:

-4	-28x	-100
x	7x²	25x
	7x	25

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 7 times -100 (a × c), and add together to equal -3 (b).

More specifically, 7 times -100 is -700. Therefore, we need to find the two numbers that multiply to equal -700, and add to equal -3.

? × ? = -700
? + ? = -3

After looking at this problem, we can see that the two numbers that multiply together to equal -700, and add together to equal -3, are -28 and 25, as illustrated here:

-28 × 25 = -700
-28 + 25 = -3

Now, we can fill in the last two squares in our box with -28x and 25x. Place -28x in the upper left square, and place 25x in the lower right square.

-4	-28x	-100
x	7x²	25x
	7x	25

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 -28x and -100. The greatest common factor of -28x and -100 is -4. Therefore, we write -4 to the left of the top row. You can see it here in the color green:

-4	-28x	-100
x	7x²	25x
	7x	25

Next, let’s look at the bottom row. We have the terms 7x² and 25x. The greatest common factor of 7x² and 25x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-4	-28x	-100
x	7x²	25x
	7x	25

To find the values below the table, we first divide 7x² by x (labeled in blue). This gives us 7x.

7x² ÷ x = 7x

You can see this value colored in orange below:

-4	-28x	-100
x	7x²	25x
	7x	25

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

25x ÷ x = 25

You can see this value colored in purple below:

-4	-28x	-100
x	7x²	25x
	7x	25

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 7x² - 3x - 100. 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:

(x - 4)(7x + 25)

That’s it! Now you know how to factor the equation 7x² - 3x - 100.

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