Factor 6x² - 35x + 51

Here we will show you how to factor the quadratic function 6x² - 35x + 51 using the box method. In other words, we will show you how to factor 6x squared minus 35x plus 51 (6x^2 - 35x + 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 6x² - 35x + 51, like this:

a = 6
b = -35
c = 51

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

-3	-18x	51
x	6x²	-17x
	6x	-17

We put 6x² (a) in the bottom left square and 51 (c) in the top right square, like this:

-3	-18x	51
x	6x²	-17x
	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 6 times 51 (a × c), and add together to equal -35 (b).

More specifically, 6 times 51 is 306. Therefore, we need to find the two numbers that multiply to equal 306, and add to equal -35.

? × ? = 306
? + ? = -35

After looking at this problem, we can see that the two numbers that multiply together to equal 306, and add together to equal -35, are -18 and -17, as illustrated here:

-18 × -17 = 306
-18 + -17 = -35

Now, we can fill in the last two squares in our box with -18x and -17x. Place -18x in the upper left square, and place -17x in the lower right square.

-3	-18x	51
x	6x²	-17x
	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 -18x and 51. The greatest common factor of -18x 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	-18x	51
x	6x²	-17x
	6x	-17

Next, let’s look at the bottom row. We have the terms 6x² and -17x. The greatest common factor of 6x² and -17x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-3	-18x	51
x	6x²	-17x
	6x	-17

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

6x² ÷ x = 6x

You can see this value colored in orange below:

-3	-18x	51
x	6x²	-17x
	6x	-17

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

-17x ÷ x = -17

You can see this value colored in purple below:

-3	-18x	51
x	6x²	-17x
	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 6x² - 35x + 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:

(x - 3)(6x - 17)

That’s it! Now you know how to factor the equation 6x² - 35x + 51.

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