Factor 100x² - 97x + 18

Here we will show you how to factor the quadratic function 100x² - 97x + 18 using the box method. In other words, we will show you how to factor 100x squared minus 97x plus 18 (100x^2 - 97x + 18) 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 + 18, like this:

a = 100
b = -97
c = 18

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

-18	-72x	18
25x	100x²	-25x
	4x	-1

We put 100x² (a) in the bottom left square and 18 (c) in the top right square, like this:

-18	-72x	18
25x	100x²	-25x
	4x	-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 100 times 18 (a × c), and add together to equal -97 (b).

More specifically, 100 times 18 is 1800. Therefore, we need to find the two numbers that multiply to equal 1800, and add to equal -97.

? × ? = 1800
? + ? = -97

After looking at this problem, we can see that the two numbers that multiply together to equal 1800, and add together to equal -97, are -72 and -25, as illustrated here:

-72 × -25 = 1800
-72 + -25 = -97

Now, we can fill in the last two squares in our box with -72x and -25x. Place -72x in the upper left square, and place -25x in the lower right square.

-18	-72x	18
25x	100x²	-25x
	4x	-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 -72x and 18. The greatest common factor of -72x and 18 is -18. Therefore, we write -18 to the left of the top row. You can see it here in the color green:

-18	-72x	18
25x	100x²	-25x
	4x	-1

Next, let’s look at the bottom row. We have the terms 100x² and -25x. The greatest common factor of 100x² and -25x is 25x. Therefore, we write 25x to the left of the bottom row. You can see it here in the color blue:

-18	-72x	18
25x	100x²	-25x
	4x	-1

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

100x² ÷ 25x = 4x

You can see this value colored in orange below:

-18	-72x	18
25x	100x²	-25x
	4x	-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:

-18	-72x	18
25x	100x²	-25x
	4x	-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 100x² - 97x + 18. 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:

(25x - 18)(4x - 1)

That’s it! Now you know how to factor the equation 100x² - 97x + 18.

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