Factor 47x² - 100x + 53

Here we will show you how to factor the quadratic function 47x² - 100x + 53 using the box method. In other words, we will show you how to factor 47x squared minus 100x plus 53 (47x^2 - 100x + 53) 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 47x² - 100x + 53, like this:

a = 47
b = -100
c = 53

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

-53	-53x	53
47x	47x²	-47x
	x	-1

We put 47x² (a) in the bottom left square and 53 (c) in the top right square, like this:

-53	-53x	53
47x	47x²	-47x
	x	-1

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 47 times 53 (a × c), and add together to equal -100 (b).

More specifically, 47 times 53 is 2491. Therefore, we need to find the two numbers that multiply to equal 2491, and add to equal -100.

? × ? = 2491
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 2491, and add together to equal -100, are -53 and -47, as illustrated here:

-53 × -47 = 2491
-53 + -47 = -100

Now, we can fill in the last two squares in our box with -53x and -47x. Place -53x in the upper left square, and place -47x in the lower right square.

-53	-53x	53
47x	47x²	-47x
	x	-1

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 -53x and 53. The greatest common factor of -53x and 53 is -53. Therefore, we write -53 to the left of the top row. You can see it here in the color green:

-53	-53x	53
47x	47x²	-47x
	x	-1

Next, let’s look at the bottom row. We have the terms 47x² and -47x. The greatest common factor of 47x² and -47x is 47x. Therefore, we write 47x to the left of the bottom row. You can see it here in the color blue:

-53	-53x	53
47x	47x²	-47x
	x	-1

To find the values below the table, we first divide 47x² by 47x (labeled in blue). This gives us x.

47x² ÷ 47x = x

You can see this value colored in orange below:

-53	-53x	53
47x	47x²	-47x
	x	-1

Next, we divide -47x by 47x (labeled in blue). This gives us -1.

-47x ÷ 47x = -1

You can see this value colored in purple below:

-53	-53x	53
47x	47x²	-47x
	x	-1

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 47x² - 100x + 53. 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:

(47x - 53)(x - 1)

That’s it! Now you know how to factor the equation 47x² - 100x + 53.

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