Factor 19x² - 100x + 25

Here we will show you how to factor the quadratic function 19x² - 100x + 25 using the box method. In other words, we will show you how to factor 19x squared minus 100x plus 25 (19x^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 19x² - 100x + 25, like this:

a = 19
b = -100
c = 25

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

-5	-95x	25
x	19x²	-5x
	19x	-5

We put 19x² (a) in the bottom left square and 25 (c) in the top right square, like this:

-5	-95x	25
x	19x²	-5x
	19x	-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 19 times 25 (a × c), and add together to equal -100 (b).

More specifically, 19 times 25 is 475. Therefore, we need to find the two numbers that multiply to equal 475, and add to equal -100.

? × ? = 475
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 475, and add together to equal -100, are -95 and -5, as illustrated here:

-95 × -5 = 475
-95 + -5 = -100

Now, we can fill in the last two squares in our box with -95x and -5x. Place -95x in the upper left square, and place -5x in the lower right square.

-5	-95x	25
x	19x²	-5x
	19x	-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 -95x and 25. The greatest common factor of -95x 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	-95x	25
x	19x²	-5x
	19x	-5

Next, let’s look at the bottom row. We have the terms 19x² and -5x. The greatest common factor of 19x² and -5x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-5	-95x	25
x	19x²	-5x
	19x	-5

To find the values below the table, we first divide 19x² by x (labeled in blue). This gives us 19x.

19x² ÷ x = 19x

You can see this value colored in orange below:

-5	-95x	25
x	19x²	-5x
	19x	-5

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

-5x ÷ x = -5

You can see this value colored in purple below:

-5	-95x	25
x	19x²	-5x
	19x	-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 19x² - 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:

(x - 5)(19x - 5)

That’s it! Now you know how to factor the equation 19x² - 100x + 25.

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