Factor 100x² - 97x

Factor 100x² - 97x - 35

Here we will show you how to factor the quadratic function 100x² - 97x - 35 using the box method. In other words, we will show you how to factor 100x squared minus 97x minus 35 (100x^2 - 97x - 35) 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 100x² - 97x - 35, like this:

a = 100
b = -97
c = -35

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

-5	-125x	-35
4x	100x²	28x
	25x	7

We put 100x² (a) in the bottom left square and -35 (c) in the top right square, like this:

-5	-125x	-35
4x	100x²	28x
	25x	7

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 100 times -35 (a × c), and add together to equal -97 (b).

More specifically, 100 times -35 is -3500. Therefore, we need to find the two numbers that multiply to equal -3500, and add to equal -97.

? × ? = -3500
? + ? = -97

After looking at this problem, we can see that the two numbers that multiply together to equal -3500, and add together to equal -97, are -125 and 28, as illustrated here:

-125 × 28 = -3500
-125 + 28 = -97

Now, we can fill in the last two squares in our box with -125x and 28x. Place -125x in the upper left square, and place 28x in the lower right square.

-5	-125x	-35
4x	100x²	28x
	25x	7

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 -125x and -35. The greatest common factor of -125x and -35 is -5. Therefore, we write -5 to the left of the top row. You can see it here in the color green:

-5	-125x	-35
4x	100x²	28x
	25x	7

Next, let’s look at the bottom row. We have the terms 100x² and 28x. The greatest common factor of 100x² and 28x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-5	-125x	-35
4x	100x²	28x
	25x	7

To find the values below the table, we first divide 100x² by 4x (labeled in blue). This gives us 25x.

100x² ÷ 4x = 25x

You can see this value colored in orange below:

-5	-125x	-35
4x	100x²	28x
	25x	7

Next, we divide 28x by 4x (labeled in blue). This gives us 7.

28x ÷ 4x = 7

You can see this value colored in purple below:

-5	-125x	-35
4x	100x²	28x
	25x	7

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 100x² - 97x - 35. 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:

(4x - 5)(25x + 7)

That’s it! Now you know how to factor the equation 100x² - 97x - 35.

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