Factor 2x² + x

Factor 2x² + x - 66

Here we will show you how to factor the quadratic function 2x² + x - 66 using the box method. In other words, we will show you how to factor 2x squared plus x minus 66 (2x^2 + x - 66) 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 2x² + x - 66, like this:

a = 2
b = 1
c = -66

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

-11	-11x	-66
2x	2x²	12x
	x	6

We put 2x² (a) in the bottom left square and -66 (c) in the top right square, like this:

-11	-11x	-66
2x	2x²	12x
	x	6

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 2 times -66 (a × c), and add together to equal 1 (b).

More specifically, 2 times -66 is -132. Therefore, we need to find the two numbers that multiply to equal -132, and add to equal 1.

? × ? = -132
? + ? = 1

After looking at this problem, we can see that the two numbers that multiply together to equal -132, and add together to equal 1, are -11 and 12, as illustrated here:

-11 × 12 = -132
-11 + 12 = 1

Now, we can fill in the last two squares in our box with -11x and 12x. Place -11x in the upper left square, and place 12x in the lower right square.

-11	-11x	-66
2x	2x²	12x
	x	6

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 -11x and -66. The greatest common factor of -11x and -66 is -11. Therefore, we write -11 to the left of the top row. You can see it here in the color green:

-11	-11x	-66
2x	2x²	12x
	x	6

Next, let’s look at the bottom row. We have the terms 2x² and 12x. The greatest common factor of 2x² and 12x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-11	-11x	-66
2x	2x²	12x
	x	6

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

2x² ÷ 2x = x

You can see this value colored in orange below:

-11	-11x	-66
2x	2x²	12x
	x	6

Next, we divide 12x by 2x (labeled in blue). This gives us 6.

12x ÷ 2x = 6

You can see this value colored in purple below:

-11	-11x	-66
2x	2x²	12x
	x	6

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 2x² + x - 66. 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 - 11)(x + 6)

That’s it! Now you know how to factor the equation 2x² + x - 66.

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