Factor 36x² - 100x + 25

Here we will show you how to factor the quadratic function 36x² - 100x + 25 using the box method. In other words, we will show you how to factor 36x squared minus 100x plus 25 (36x^2 - 100x + 25) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 36x² - 100x + 25, like this:

a = 36
b = -100
c = 25

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

-5	-90x	25
2x	36x²	-10x
	18x	-5

We put 36x² (a) in the bottom left square and 25 (c) in the top right square, like this:

-5	-90x	25
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 25 (a × c), and add together to equal -100 (b).

More specifically, 36 times 25 is 900. Therefore, we need to find the two numbers that multiply to equal 900, and add to equal -100.

? × ? = 900
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 900, and add together to equal -100, are -90 and -10, as illustrated here:

-90 × -10 = 900
-90 + -10 = -100

Now, we can fill in the last two squares in our box with -90x and -10x. Place -90x in the upper left square, and place -10x in the lower right square.

-5	-90x	25
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 -90x and 25. The greatest common factor of -90x and 25 is -5. Therefore, we write -5 to the left of the top row. You can see it here in the color green:

-5	-90x	25
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:

-5	-90x	25
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:

-5	-90x	25
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:

-5	-90x	25
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 - 100x + 25. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

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

That’s it! Now you know how to factor the equation 36x² - 100x + 25.

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