Factor 18x² + 75x + 68

Here we will show you how to factor the quadratic function 18x² + 75x + 68 using the box method. In other words, we will show you how to factor 18x squared plus 75x plus 68 (18x^2 + 75x + 68) 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 + 75x + 68, like this:

a = 18
b = 75
c = 68

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

4	24x	68
3x	18x²	51x
	6x	17

We put 18x² (a) in the bottom left square and 68 (c) in the top right square, like this:

4	24x	68
3x	18x²	51x
	6x	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 68 (a × c), and add together to equal 75 (b).

More specifically, 18 times 68 is 1224. Therefore, we need to find the two numbers that multiply to equal 1224, and add to equal 75.

? × ? = 1224
? + ? = 75

After looking at this problem, we can see that the two numbers that multiply together to equal 1224, and add together to equal 75, are 24 and 51, as illustrated here:

24 × 51 = 1224
24 + 51 = 75

Now, we can fill in the last two squares in our box with 24x and 51x. Place 24x in the upper left square, and place 51x in the lower right square.

4	24x	68
3x	18x²	51x
	6x	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 24x and 68. The greatest common factor of 24x and 68 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	24x	68
3x	18x²	51x
	6x	17

Next, let’s look at the bottom row. We have the terms 18x² and 51x. The greatest common factor of 18x² and 51x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

4	24x	68
3x	18x²	51x
	6x	17

To find the values below the table, we first divide 18x² by 3x (labeled in blue). This gives us 6x.

18x² ÷ 3x = 6x

You can see this value colored in orange below:

4	24x	68
3x	18x²	51x
	6x	17

Next, we divide 51x by 3x (labeled in blue). This gives us 17.

51x ÷ 3x = 17

You can see this value colored in purple below:

4	24x	68
3x	18x²	51x
	6x	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 + 75x + 68. 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:

(3x + 4)(6x + 17)

That’s it! Now you know how to factor the equation 18x² + 75x + 68.

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