Factor -36x² - 73x

Factor -36x² - 73x - 35

Here we will show you how to factor the quadratic function -36x² - 73x - 35 using the box method. In other words, we will show you how to factor negative 36x squared minus 73x minus 35 (-36x^2 - 73x - 35) 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 + 73x + 35. Now we can label the different parts of our equation, like this:

a = 36
b = 73
c = 35

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

7	28x	35
9x	36x²	45x
	4x	5

We put 36x² (a) in the bottom left square and 35 (c) in the top right square, like this:

7	28x	35
9x	36x²	45x
	4x	5

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 35 (a × c), and add together to equal 73 (b).

More specifically, 36 times 35 is 1260. Therefore, we need to find the two numbers that multiply to equal 1260, and add to equal 73.

? × ? = 1260
? + ? = 73

After looking at this problem, we can see that the two numbers that multiply together to equal 1260, and add together to equal 73, are 28 and 45, as illustrated here:

28 × 45 = 1260
28 + 45 = 73

Now, we can fill in the last two squares in our box with 28x and 45x. Place 28x in the upper left square, and place 45x in the lower right square.

7	28x	35
9x	36x²	45x
	4x	5

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 28x and 35. The greatest common factor of 28x and 35 is 7. Therefore, we write 7 to the left of the top row. You can see it here in the color green:

7	28x	35
9x	36x²	45x
	4x	5

Next, let’s look at the bottom row. We have the terms 36x² and 45x. The greatest common factor of 36x² and 45x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

7	28x	35
9x	36x²	45x
	4x	5

To find the values below the table, we first divide 36x² by 9x (labeled in blue). This gives us 4x.

36x² ÷ 9x = 4x

You can see this value colored in orange below:

7	28x	35
9x	36x²	45x
	4x	5

Next, we divide 45x by 9x (labeled in blue). This gives us 5.

45x ÷ 9x = 5

You can see this value colored in purple below:

7	28x	35
9x	36x²	45x
	4x	5

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 - 73x - 35. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(9x + 7)(4x + 5)

In our original quadratic equation, -36x² - 73x - 35, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(9x + 7)(4x + 5)

That’s it! Now you know how to factor the equation -36x² - 73x - 35.

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