Factor 18x² - 37x + 15

Here we will show you how to factor the quadratic function 18x² - 37x + 15 using the box method. In other words, we will show you how to factor 18x squared minus 37x plus 15 (18x^2 - 37x + 15) 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 18x² - 37x + 15, like this:

a = 18
b = -37
c = 15

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

-3	-27x	15
2x	18x²	-10x
	9x	-5

We put 18x² (a) in the bottom left square and 15 (c) in the top right square, like this:

-3	-27x	15
2x	18x²	-10x
	9x	-5

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 18 times 15 (a × c), and add together to equal -37 (b).

More specifically, 18 times 15 is 270. Therefore, we need to find the two numbers that multiply to equal 270, and add to equal -37.

? × ? = 270
? + ? = -37

After looking at this problem, we can see that the two numbers that multiply together to equal 270, and add together to equal -37, are -27 and -10, as illustrated here:

-27 × -10 = 270
-27 + -10 = -37

Now, we can fill in the last two squares in our box with -27x and -10x. Place -27x in the upper left square, and place -10x in the lower right square.

-3	-27x	15
2x	18x²	-10x
	9x	-5

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 -27x and 15. The greatest common factor of -27x and 15 is -3. Therefore, we write -3 to the left of the top row. You can see it here in the color green:

-3	-27x	15
2x	18x²	-10x
	9x	-5

Next, let’s look at the bottom row. We have the terms 18x² and -10x. The greatest common factor of 18x² and -10x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-3	-27x	15
2x	18x²	-10x
	9x	-5

To find the values below the table, we first divide 18x² by 2x (labeled in blue). This gives us 9x.

18x² ÷ 2x = 9x

You can see this value colored in orange below:

-3	-27x	15
2x	18x²	-10x
	9x	-5

Next, we divide -10x by 2x (labeled in blue). This gives us -5.

-10x ÷ 2x = -5

You can see this value colored in purple below:

-3	-27x	15
2x	18x²	-10x
	9x	-5

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 18x² - 37x + 15. 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:

(2x - 3)(9x - 5)

That’s it! Now you know how to factor the equation 18x² - 37x + 15.

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