Factor -36x² - 76x

Factor -36x² - 76x - 29

Here we will show you how to factor the quadratic function -36x² - 76x - 29 using the box method. In other words, we will show you how to factor negative 36x squared minus 76x minus 29 (-36x^2 - 76x - 29) 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 36x² + 76x + 29. Now we can label the different parts of our equation, like this:

a = 36
b = 76
c = 29

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

1	18x	29
2x	36x²	58x
	18x	29

We put 36x² (a) in the bottom left square and 29 (c) in the top right square, like this:

1	18x	29
2x	36x²	58x
	18x	29

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 36 times 29 (a × c), and add together to equal 76 (b).

More specifically, 36 times 29 is 1044. Therefore, we need to find the two numbers that multiply to equal 1044, and add to equal 76.

? × ? = 1044
? + ? = 76

After looking at this problem, we can see that the two numbers that multiply together to equal 1044, and add together to equal 76, are 18 and 58, as illustrated here:

18 × 58 = 1044
18 + 58 = 76

Now, we can fill in the last two squares in our box with 18x and 58x. Place 18x in the upper left square, and place 58x in the lower right square.

1	18x	29
2x	36x²	58x
	18x	29

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 18x and 29. The greatest common factor of 18x and 29 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	18x	29
2x	36x²	58x
	18x	29

Next, let’s look at the bottom row. We have the terms 36x² and 58x. The greatest common factor of 36x² and 58x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	18x	29
2x	36x²	58x
	18x	29

To find the values below the table, we first divide 36x² by 2x (labeled in blue). This gives us 18x.

36x² ÷ 2x = 18x

You can see this value colored in orange below:

1	18x	29
2x	36x²	58x
	18x	29

Next, we divide 58x by 2x (labeled in blue). This gives us 29.

58x ÷ 2x = 29

You can see this value colored in purple below:

1	18x	29
2x	36x²	58x
	18x	29

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 -36x² - 76x - 29. 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)(18x + 29)

In our original quadratic equation, -36x² - 76x - 29, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x + 1)(18x + 29)

That’s it! Now you know how to factor the equation -36x² - 76x - 29.

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