Factor 18x² + 61x + 51

Here we will show you how to factor the quadratic function 18x² + 61x + 51 using the box method. In other words, we will show you how to factor 18x squared plus 61x plus 51 (18x^2 + 61x + 51) 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 + 61x + 51, like this:

a = 18
b = 61
c = 51

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

3	27x	51
2x	18x²	34x
	9x	17

We put 18x² (a) in the bottom left square and 51 (c) in the top right square, like this:

3	27x	51
2x	18x²	34x
	9x	17

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 51 (a × c), and add together to equal 61 (b).

More specifically, 18 times 51 is 918. Therefore, we need to find the two numbers that multiply to equal 918, and add to equal 61.

? × ? = 918
? + ? = 61

After looking at this problem, we can see that the two numbers that multiply together to equal 918, and add together to equal 61, are 27 and 34, as illustrated here:

27 × 34 = 918
27 + 34 = 61

Now, we can fill in the last two squares in our box with 27x and 34x. Place 27x in the upper left square, and place 34x in the lower right square.

3	27x	51
2x	18x²	34x
	9x	17

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 27x and 51. The greatest common factor of 27x and 51 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	27x	51
2x	18x²	34x
	9x	17

Next, let’s look at the bottom row. We have the terms 18x² and 34x. The greatest common factor of 18x² and 34x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

3	27x	51
2x	18x²	34x
	9x	17

To find the values below the table, we first divide 18x² by 2x (labeled in blue). This gives us 9x.

18x² ÷ 2x = 9x

You can see this value colored in orange below:

3	27x	51
2x	18x²	34x
	9x	17

Next, we divide 34x by 2x (labeled in blue). This gives us 17.

34x ÷ 2x = 17

You can see this value colored in purple below:

3	27x	51
2x	18x²	34x
	9x	17

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 + 61x + 51. 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:

(2x + 3)(9x + 17)

That’s it! Now you know how to factor the equation 18x² + 61x + 51.

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