Factor 37x² - 100x + 63

Here we will show you how to factor the quadratic function 37x² - 100x + 63 using the box method. In other words, we will show you how to factor 37x squared minus 100x plus 63 (37x^2 - 100x + 63) 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 37x² - 100x + 63, like this:

a = 37
b = -100
c = 63

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

-63	-63x	63
37x	37x²	-37x
	x	-1

We put 37x² (a) in the bottom left square and 63 (c) in the top right square, like this:

-63	-63x	63
37x	37x²	-37x
	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 37 times 63 (a × c), and add together to equal -100 (b).

More specifically, 37 times 63 is 2331. Therefore, we need to find the two numbers that multiply to equal 2331, and add to equal -100.

? × ? = 2331
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 2331, and add together to equal -100, are -63 and -37, as illustrated here:

-63 × -37 = 2331
-63 + -37 = -100

Now, we can fill in the last two squares in our box with -63x and -37x. Place -63x in the upper left square, and place -37x in the lower right square.

-63	-63x	63
37x	37x²	-37x
	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 -63x and 63. The greatest common factor of -63x and 63 is -63. Therefore, we write -63 to the left of the top row. You can see it here in the color green:

-63	-63x	63
37x	37x²	-37x
	x	-1

Next, let’s look at the bottom row. We have the terms 37x² and -37x. The greatest common factor of 37x² and -37x is 37x. Therefore, we write 37x to the left of the bottom row. You can see it here in the color blue:

-63	-63x	63
37x	37x²	-37x
	x	-1

To find the values below the table, we first divide 37x² by 37x (labeled in blue). This gives us x.

37x² ÷ 37x = x

You can see this value colored in orange below:

-63	-63x	63
37x	37x²	-37x
	x	-1

Next, we divide -37x by 37x (labeled in blue). This gives us -1.

-37x ÷ 37x = -1

You can see this value colored in purple below:

-63	-63x	63
37x	37x²	-37x
	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 37x² - 100x + 63. 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:

(37x - 63)(x - 1)

That’s it! Now you know how to factor the equation 37x² - 100x + 63.

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