Factor 17x² - 50x + 33

Here we will show you how to factor the quadratic function 17x² - 50x + 33 using the box method. In other words, we will show you how to factor 17x squared minus 50x plus 33 (17x^2 - 50x + 33) 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 17x² - 50x + 33, like this:

a = 17
b = -50
c = 33

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

-33	-33x	33
17x	17x²	-17x
	x	-1

We put 17x² (a) in the bottom left square and 33 (c) in the top right square, like this:

-33	-33x	33
17x	17x²	-17x
	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 17 times 33 (a × c), and add together to equal -50 (b).

More specifically, 17 times 33 is 561. Therefore, we need to find the two numbers that multiply to equal 561, and add to equal -50.

? × ? = 561
? + ? = -50

After looking at this problem, we can see that the two numbers that multiply together to equal 561, and add together to equal -50, are -33 and -17, as illustrated here:

-33 × -17 = 561
-33 + -17 = -50

Now, we can fill in the last two squares in our box with -33x and -17x. Place -33x in the upper left square, and place -17x in the lower right square.

-33	-33x	33
17x	17x²	-17x
	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 -33x and 33. The greatest common factor of -33x and 33 is -33. Therefore, we write -33 to the left of the top row. You can see it here in the color green:

-33	-33x	33
17x	17x²	-17x
	x	-1

Next, let’s look at the bottom row. We have the terms 17x² and -17x. The greatest common factor of 17x² and -17x is 17x. Therefore, we write 17x to the left of the bottom row. You can see it here in the color blue:

-33	-33x	33
17x	17x²	-17x
	x	-1

To find the values below the table, we first divide 17x² by 17x (labeled in blue). This gives us x.

17x² ÷ 17x = x

You can see this value colored in orange below:

-33	-33x	33
17x	17x²	-17x
	x	-1

Next, we divide -17x by 17x (labeled in blue). This gives us -1.

-17x ÷ 17x = -1

You can see this value colored in purple below:

-33	-33x	33
17x	17x²	-17x
	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 17x² - 50x + 33. 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:

(17x - 33)(x - 1)

That’s it! Now you know how to factor the equation 17x² - 50x + 33.

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