Factor 15x² + 6x

Factor 15x² + 6x - 9

Here we will show you how to factor the quadratic function 15x² + 6x - 9 using the box method. In other words, we will show you how to factor 15x squared plus 6x minus 9 (15x^2 + 6x - 9) 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 + 6x - 9, like this:

a = 15
b = 6
c = -9

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

-9	-9x	-9
15x	15x²	15x
	x	1

We put 15x² (a) in the bottom left square and -9 (c) in the top right square, like this:

-9	-9x	-9
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 -9 (a × c), and add together to equal 6 (b).

More specifically, 15 times -9 is -135. Therefore, we need to find the two numbers that multiply to equal -135, and add to equal 6.

? × ? = -135
? + ? = 6

After looking at this problem, we can see that the two numbers that multiply together to equal -135, and add together to equal 6, are -9 and 15, as illustrated here:

-9 × 15 = -135
-9 + 15 = 6

Now, we can fill in the last two squares in our box with -9x and 15x. Place -9x in the upper left square, and place 15x in the lower right square.

-9	-9x	-9
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 -9x and -9. The greatest common factor of -9x and -9 is -9. Therefore, we write -9 to the left of the top row. You can see it here in the color green:

-9	-9x	-9
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:

-9	-9x	-9
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:

-9	-9x	-9
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:

-9	-9x	-9
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 + 6x - 9. 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 - 9)(x + 1)

That’s it! Now you know how to factor the equation 15x² + 6x - 9.

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