Factor -36x² + 28x

Factor -36x² + 28x - 5

Here we will show you how to factor the quadratic function -36x² + 28x - 5 using the box method. In other words, we will show you how to factor negative 36x squared plus 28x minus 5 (-36x^2 + 28x - 5) 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 - 28x + 5. Now we can label the different parts of our equation, like this:

a = 36
b = -28
c = 5

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

-1	-18x	5
2x	36x²	-10x
	18x	-5

We put 36x² (a) in the bottom left square and 5 (c) in the top right square, like this:

-1	-18x	5
2x	36x²	-10x
	18x	-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 5 (a × c), and add together to equal -28 (b).

More specifically, 36 times 5 is 180. Therefore, we need to find the two numbers that multiply to equal 180, and add to equal -28.

? × ? = 180
? + ? = -28

After looking at this problem, we can see that the two numbers that multiply together to equal 180, and add together to equal -28, are -18 and -10, as illustrated here:

-18 × -10 = 180
-18 + -10 = -28

Now, we can fill in the last two squares in our box with -18x and -10x. Place -18x in the upper left square, and place -10x in the lower right square.

-1	-18x	5
2x	36x²	-10x
	18x	-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 -18x and 5. The greatest common factor of -18x and 5 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-18x	5
2x	36x²	-10x
	18x	-5

Next, let’s look at the bottom row. We have the terms 36x² and -10x. The greatest common factor of 36x² and -10x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-1	-18x	5
2x	36x²	-10x
	18x	-5

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	5
2x	36x²	-10x
	18x	-5

Next, we divide -10x by 2x (labeled in blue). This gives us -5.

-10x ÷ 2x = -5

You can see this value colored in purple below:

-1	-18x	5
2x	36x²	-10x
	18x	-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 + 28x - 5. 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 - 5)

In our original quadratic equation, -36x² + 28x - 5, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x - 1)(18x - 5)

That’s it! Now you know how to factor the equation -36x² + 28x - 5.

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