Factor 15x² + 7x

Factor 15x² + 7x - 22

Here we will show you how to factor the quadratic function 15x² + 7x - 22 using the box method. In other words, we will show you how to factor 15x squared plus 7x minus 22 (15x^2 + 7x - 22) 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 + 7x - 22, like this:

a = 15
b = 7
c = -22

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

-1	-15x	-22
x	15x²	22x
	15x	22

We put 15x² (a) in the bottom left square and -22 (c) in the top right square, like this:

-1	-15x	-22
x	15x²	22x
	15x	22

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 -22 (a × c), and add together to equal 7 (b).

More specifically, 15 times -22 is -330. Therefore, we need to find the two numbers that multiply to equal -330, and add to equal 7.

? × ? = -330
? + ? = 7

After looking at this problem, we can see that the two numbers that multiply together to equal -330, and add together to equal 7, are -15 and 22, as illustrated here:

-15 × 22 = -330
-15 + 22 = 7

Now, we can fill in the last two squares in our box with -15x and 22x. Place -15x in the upper left square, and place 22x in the lower right square.

-1	-15x	-22
x	15x²	22x
	15x	22

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 -15x and -22. The greatest common factor of -15x and -22 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-15x	-22
x	15x²	22x
	15x	22

Next, let’s look at the bottom row. We have the terms 15x² and 22x. The greatest common factor of 15x² and 22x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-15x	-22
x	15x²	22x
	15x	22

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

15x² ÷ x = 15x

You can see this value colored in orange below:

-1	-15x	-22
x	15x²	22x
	15x	22

Next, we divide 22x by x (labeled in blue). This gives us 22.

22x ÷ x = 22

You can see this value colored in purple below:

-1	-15x	-22
x	15x²	22x
	15x	22

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 + 7x - 22. 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 - 1)(15x + 22)

That’s it! Now you know how to factor the equation 15x² + 7x - 22.

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