Factor -34x² - 61x

Factor -34x² - 61x - 22

Here we will show you how to factor the quadratic function -34x² - 61x - 22 using the box method. In other words, we will show you how to factor negative 34x squared minus 61x minus 22 (-34x^2 - 61x - 22) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 34x² + 61x + 22. Now we can label the different parts of our equation, like this:

a = 34
b = 61
c = 22

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

1	17x	22
2x	34x²	44x
	17x	22

We put 34x² (a) in the bottom left square and 22 (c) in the top right square, like this:

1	17x	22
2x	34x²	44x
	17x	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 34 times 22 (a × c), and add together to equal 61 (b).

More specifically, 34 times 22 is 748. Therefore, we need to find the two numbers that multiply to equal 748, and add to equal 61.

? × ? = 748
? + ? = 61

After looking at this problem, we can see that the two numbers that multiply together to equal 748, and add together to equal 61, are 17 and 44, as illustrated here:

17 × 44 = 748
17 + 44 = 61

Now, we can fill in the last two squares in our box with 17x and 44x. Place 17x in the upper left square, and place 44x in the lower right square.

1	17x	22
2x	34x²	44x
	17x	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 17x and 22. The greatest common factor of 17x 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	17x	22
2x	34x²	44x
	17x	22

Next, let’s look at the bottom row. We have the terms 34x² and 44x. The greatest common factor of 34x² and 44x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	17x	22
2x	34x²	44x
	17x	22

To find the values below the table, we first divide 34x² by 2x (labeled in blue). This gives us 17x.

34x² ÷ 2x = 17x

You can see this value colored in orange below:

1	17x	22
2x	34x²	44x
	17x	22

Next, we divide 44x by 2x (labeled in blue). This gives us 22.

44x ÷ 2x = 22

You can see this value colored in purple below:

1	17x	22
2x	34x²	44x
	17x	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 -34x² - 61x - 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:

(2x + 1)(17x + 22)

In our original quadratic equation, -34x² - 61x - 22, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x + 1)(17x + 22)

That’s it! Now you know how to factor the equation -34x² - 61x - 22.

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