Factor 9x² - 68x + 100

Here we will show you how to factor the quadratic function 9x² - 68x + 100 using the box method. In other words, we will show you how to factor 9x squared minus 68x plus 100 (9x^2 - 68x + 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 9x² - 68x + 100, like this:

a = 9
b = -68
c = 100

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

-50	-50x	100
9x	9x²	-18x
	x	-2

We put 9x² (a) in the bottom left square and 100 (c) in the top right square, like this:

-50	-50x	100
9x	9x²	-18x
	x	-2

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 9 times 100 (a × c), and add together to equal -68 (b).

More specifically, 9 times 100 is 900. Therefore, we need to find the two numbers that multiply to equal 900, and add to equal -68.

? × ? = 900
? + ? = -68

After looking at this problem, we can see that the two numbers that multiply together to equal 900, and add together to equal -68, are -50 and -18, as illustrated here:

-50 × -18 = 900
-50 + -18 = -68

Now, we can fill in the last two squares in our box with -50x and -18x. Place -50x in the upper left square, and place -18x in the lower right square.

-50	-50x	100
9x	9x²	-18x
	x	-2

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 -50x and 100. The greatest common factor of -50x and 100 is -50. Therefore, we write -50 to the left of the top row. You can see it here in the color green:

-50	-50x	100
9x	9x²	-18x
	x	-2

Next, let’s look at the bottom row. We have the terms 9x² and -18x. The greatest common factor of 9x² and -18x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

-50	-50x	100
9x	9x²	-18x
	x	-2

To find the values below the table, we first divide 9x² by 9x (labeled in blue). This gives us x.

9x² ÷ 9x = x

You can see this value colored in orange below:

-50	-50x	100
9x	9x²	-18x
	x	-2

Next, we divide -18x by 9x (labeled in blue). This gives us -2.

-18x ÷ 9x = -2

You can see this value colored in purple below:

-50	-50x	100
9x	9x²	-18x
	x	-2

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 9x² - 68x + 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:

(9x - 50)(x - 2)

That’s it! Now you know how to factor the equation 9x² - 68x + 100.

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