Factor 9x² - 68x + 84

Here we will show you how to factor the quadratic function 9x² - 68x + 84 using the box method. In other words, we will show you how to factor 9x squared minus 68x plus 84 (9x^2 - 68x + 84) 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 + 84, like this:

a = 9
b = -68
c = 84

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

-6	-54x	84
x	9x²	-14x
	9x	-14

We put 9x² (a) in the bottom left square and 84 (c) in the top right square, like this:

-6	-54x	84
x	9x²	-14x
	9x	-14

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 84 (a × c), and add together to equal -68 (b).

More specifically, 9 times 84 is 756. Therefore, we need to find the two numbers that multiply to equal 756, and add to equal -68.

? × ? = 756
? + ? = -68

After looking at this problem, we can see that the two numbers that multiply together to equal 756, and add together to equal -68, are -54 and -14, as illustrated here:

-54 × -14 = 756
-54 + -14 = -68

Now, we can fill in the last two squares in our box with -54x and -14x. Place -54x in the upper left square, and place -14x in the lower right square.

-6	-54x	84
x	9x²	-14x
	9x	-14

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 -54x and 84. The greatest common factor of -54x and 84 is -6. Therefore, we write -6 to the left of the top row. You can see it here in the color green:

-6	-54x	84
x	9x²	-14x
	9x	-14

Next, let’s look at the bottom row. We have the terms 9x² and -14x. The greatest common factor of 9x² and -14x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-6	-54x	84
x	9x²	-14x
	9x	-14

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

9x² ÷ x = 9x

You can see this value colored in orange below:

-6	-54x	84
x	9x²	-14x
	9x	-14

Next, we divide -14x by x (labeled in blue). This gives us -14.

-14x ÷ x = -14

You can see this value colored in purple below:

-6	-54x	84
x	9x²	-14x
	9x	-14

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 + 84. 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 - 6)(9x - 14)

That’s it! Now you know how to factor the equation 9x² - 68x + 84.

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