Factor -34x² - 61x

Factor -34x² - 61x - 27

Here we will show you how to factor the quadratic function -34x² - 61x - 27 using the box method. In other words, we will show you how to factor negative 34x squared minus 61x minus 27 (-34x^2 - 61x - 27) 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 + 27. Now we can label the different parts of our equation, like this:

a = 34
b = 61
c = 27

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

27	27x	27
34x	34x²	34x
	x	1

We put 34x² (a) in the bottom left square and 27 (c) in the top right square, like this:

27	27x	27
34x	34x²	34x
	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 34 times 27 (a × c), and add together to equal 61 (b).

More specifically, 34 times 27 is 918. Therefore, we need to find the two numbers that multiply to equal 918, and add to equal 61.

? × ? = 918
? + ? = 61

After looking at this problem, we can see that the two numbers that multiply together to equal 918, and add together to equal 61, are 27 and 34, as illustrated here:

27 × 34 = 918
27 + 34 = 61

Now, we can fill in the last two squares in our box with 27x and 34x. Place 27x in the upper left square, and place 34x in the lower right square.

27	27x	27
34x	34x²	34x
	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 27x and 27. The greatest common factor of 27x and 27 is 27. Therefore, we write 27 to the left of the top row. You can see it here in the color green:

27	27x	27
34x	34x²	34x
	x	1

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

27	27x	27
34x	34x²	34x
	x	1

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

34x² ÷ 34x = x

You can see this value colored in orange below:

27	27x	27
34x	34x²	34x
	x	1

Next, we divide 34x by 34x (labeled in blue). This gives us 1.

34x ÷ 34x = 1

You can see this value colored in purple below:

27	27x	27
34x	34x²	34x
	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 -34x² - 61x - 27. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(34x + 27)(x + 1)

In our original quadratic equation, -34x² - 61x - 27, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(34x + 27)(x + 1)

That’s it! Now you know how to factor the equation -34x² - 61x - 27.

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