Factor 15x² - 69x + 54

Here we will show you how to factor the quadratic function 15x² - 69x + 54 using the box method. In other words, we will show you how to factor 15x squared minus 69x plus 54 (15x^2 - 69x + 54) 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 15x² - 69x + 54, like this:

a = 15
b = -69
c = 54

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

-54	-54x	54
15x	15x²	-15x
	x	-1

We put 15x² (a) in the bottom left square and 54 (c) in the top right square, like this:

-54	-54x	54
15x	15x²	-15x
	x	-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 15 times 54 (a × c), and add together to equal -69 (b).

More specifically, 15 times 54 is 810. Therefore, we need to find the two numbers that multiply to equal 810, and add to equal -69.

? × ? = 810
? + ? = -69

After looking at this problem, we can see that the two numbers that multiply together to equal 810, and add together to equal -69, are -54 and -15, as illustrated here:

-54 × -15 = 810
-54 + -15 = -69

Now, we can fill in the last two squares in our box with -54x and -15x. Place -54x in the upper left square, and place -15x in the lower right square.

-54	-54x	54
15x	15x²	-15x
	x	-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 -54x and 54. The greatest common factor of -54x and 54 is -54. Therefore, we write -54 to the left of the top row. You can see it here in the color green:

-54	-54x	54
15x	15x²	-15x
	x	-1

Next, let’s look at the bottom row. We have the terms 15x² and -15x. The greatest common factor of 15x² and -15x is 15x. Therefore, we write 15x to the left of the bottom row. You can see it here in the color blue:

-54	-54x	54
15x	15x²	-15x
	x	-1

To find the values below the table, we first divide 15x² by 15x (labeled in blue). This gives us x.

15x² ÷ 15x = x

You can see this value colored in orange below:

-54	-54x	54
15x	15x²	-15x
	x	-1

Next, we divide -15x by 15x (labeled in blue). This gives us -1.

-15x ÷ 15x = -1

You can see this value colored in purple below:

-54	-54x	54
15x	15x²	-15x
	x	-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 15x² - 69x + 54. 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:

(15x - 54)(x - 1)

That’s it! Now you know how to factor the equation 15x² - 69x + 54.

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