Factor -34x² - 65x

Factor -34x² - 65x - 21

Here we will show you how to factor the quadratic function -34x² - 65x - 21 using the box method. In other words, we will show you how to factor negative 34x squared minus 65x minus 21 (-34x^2 - 65x - 21) 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 + 65x + 21. Now we can label the different parts of our equation, like this:

a = 34
b = 65
c = 21

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

7	14x	21
17x	34x²	51x
	2x	3

We put 34x² (a) in the bottom left square and 21 (c) in the top right square, like this:

7	14x	21
17x	34x²	51x
	2x	3

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 21 (a × c), and add together to equal 65 (b).

More specifically, 34 times 21 is 714. Therefore, we need to find the two numbers that multiply to equal 714, and add to equal 65.

? × ? = 714
? + ? = 65

After looking at this problem, we can see that the two numbers that multiply together to equal 714, and add together to equal 65, are 14 and 51, as illustrated here:

14 × 51 = 714
14 + 51 = 65

Now, we can fill in the last two squares in our box with 14x and 51x. Place 14x in the upper left square, and place 51x in the lower right square.

7	14x	21
17x	34x²	51x
	2x	3

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 14x and 21. The greatest common factor of 14x and 21 is 7. Therefore, we write 7 to the left of the top row. You can see it here in the color green:

7	14x	21
17x	34x²	51x
	2x	3

Next, let’s look at the bottom row. We have the terms 34x² and 51x. The greatest common factor of 34x² and 51x is 17x. Therefore, we write 17x to the left of the bottom row. You can see it here in the color blue:

7	14x	21
17x	34x²	51x
	2x	3

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

34x² ÷ 17x = 2x

You can see this value colored in orange below:

7	14x	21
17x	34x²	51x
	2x	3

Next, we divide 51x by 17x (labeled in blue). This gives us 3.

51x ÷ 17x = 3

You can see this value colored in purple below:

7	14x	21
17x	34x²	51x
	2x	3

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 - 65x - 21. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(17x + 7)(2x + 3)

In our original quadratic equation, -34x² - 65x - 21, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(17x + 7)(2x + 3)

That’s it! Now you know how to factor the equation -34x² - 65x - 21.

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