Factor 61x² + 94x + 33

Here we will show you how to factor the quadratic function 61x² + 94x + 33 using the box method. In other words, we will show you how to factor 61x squared plus 94x plus 33 (61x^2 + 94x + 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 61x² + 94x + 33, like this:

a = 61
b = 94
c = 33

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

33	33x	33
61x	61x²	61x
	x	1

We put 61x² (a) in the bottom left square and 33 (c) in the top right square, like this:

33	33x	33
61x	61x²	61x
	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 61 times 33 (a × c), and add together to equal 94 (b).

More specifically, 61 times 33 is 2013. Therefore, we need to find the two numbers that multiply to equal 2013, and add to equal 94.

? × ? = 2013
? + ? = 94

After looking at this problem, we can see that the two numbers that multiply together to equal 2013, and add together to equal 94, are 33 and 61, as illustrated here:

33 × 61 = 2013
33 + 61 = 94

Now, we can fill in the last two squares in our box with 33x and 61x. Place 33x in the upper left square, and place 61x in the lower right square.

33	33x	33
61x	61x²	61x
	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
61x	61x²	61x
	x	1

Next, let’s look at the bottom row. We have the terms 61x² and 61x. The greatest common factor of 61x² and 61x is 61x. Therefore, we write 61x to the left of the bottom row. You can see it here in the color blue:

33	33x	33
61x	61x²	61x
	x	1

To find the values below the table, we first divide 61x² by 61x (labeled in blue). This gives us x.

61x² ÷ 61x = x

You can see this value colored in orange below:

33	33x	33
61x	61x²	61x
	x	1

Next, we divide 61x by 61x (labeled in blue). This gives us 1.

61x ÷ 61x = 1

You can see this value colored in purple below:

33	33x	33
61x	61x²	61x
	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 61x² + 94x + 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:

(61x + 33)(x + 1)

That’s it! Now you know how to factor the equation 61x² + 94x + 33.

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