Factor 18x² + 85x + 98

Here we will show you how to factor the quadratic function 18x² + 85x + 98 using the box method. In other words, we will show you how to factor 18x squared plus 85x plus 98 (18x^2 + 85x + 98) 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 18x² + 85x + 98, like this:

a = 18
b = 85
c = 98

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

2	36x	98
x	18x²	49x
	18x	49

We put 18x² (a) in the bottom left square and 98 (c) in the top right square, like this:

2	36x	98
x	18x²	49x
	18x	49

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 18 times 98 (a × c), and add together to equal 85 (b).

More specifically, 18 times 98 is 1764. Therefore, we need to find the two numbers that multiply to equal 1764, and add to equal 85.

? × ? = 1764
? + ? = 85

After looking at this problem, we can see that the two numbers that multiply together to equal 1764, and add together to equal 85, are 36 and 49, as illustrated here:

36 × 49 = 1764
36 + 49 = 85

Now, we can fill in the last two squares in our box with 36x and 49x. Place 36x in the upper left square, and place 49x in the lower right square.

2	36x	98
x	18x²	49x
	18x	49

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 36x and 98. The greatest common factor of 36x and 98 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	36x	98
x	18x²	49x
	18x	49

Next, let’s look at the bottom row. We have the terms 18x² and 49x. The greatest common factor of 18x² and 49x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

2	36x	98
x	18x²	49x
	18x	49

To find the values below the table, we first divide 18x² by x (labeled in blue). This gives us 18x.

18x² ÷ x = 18x

You can see this value colored in orange below:

2	36x	98
x	18x²	49x
	18x	49

Next, we divide 49x by x (labeled in blue). This gives us 49.

49x ÷ x = 49

You can see this value colored in purple below:

2	36x	98
x	18x²	49x
	18x	49

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 18x² + 85x + 98. 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 + 2)(18x + 49)

That’s it! Now you know how to factor the equation 18x² + 85x + 98.

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Factor 18x² + 85x + 100
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