Factor 8x² - 59x + 51

Here we will show you how to factor the quadratic function 8x² - 59x + 51 using the box method. In other words, we will show you how to factor 8x squared minus 59x plus 51 (8x^2 - 59x + 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 8x² - 59x + 51, like this:

a = 8
b = -59
c = 51

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

-51	-51x	51
8x	8x²	-8x
	x	-1

We put 8x² (a) in the bottom left square and 51 (c) in the top right square, like this:

-51	-51x	51
8x	8x²	-8x
	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 8 times 51 (a × c), and add together to equal -59 (b).

More specifically, 8 times 51 is 408. Therefore, we need to find the two numbers that multiply to equal 408, and add to equal -59.

? × ? = 408
? + ? = -59

After looking at this problem, we can see that the two numbers that multiply together to equal 408, and add together to equal -59, are -51 and -8, as illustrated here:

-51 × -8 = 408
-51 + -8 = -59

Now, we can fill in the last two squares in our box with -51x and -8x. Place -51x in the upper left square, and place -8x in the lower right square.

-51	-51x	51
8x	8x²	-8x
	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 -51x and 51. The greatest common factor of -51x and 51 is -51. Therefore, we write -51 to the left of the top row. You can see it here in the color green:

-51	-51x	51
8x	8x²	-8x
	x	-1

Next, let’s look at the bottom row. We have the terms 8x² and -8x. The greatest common factor of 8x² and -8x is 8x. Therefore, we write 8x to the left of the bottom row. You can see it here in the color blue:

-51	-51x	51
8x	8x²	-8x
	x	-1

To find the values below the table, we first divide 8x² by 8x (labeled in blue). This gives us x.

8x² ÷ 8x = x

You can see this value colored in orange below:

-51	-51x	51
8x	8x²	-8x
	x	-1

Next, we divide -8x by 8x (labeled in blue). This gives us -1.

-8x ÷ 8x = -1

You can see this value colored in purple below:

-51	-51x	51
8x	8x²	-8x
	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 8x² - 59x + 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:

(8x - 51)(x - 1)

That’s it! Now you know how to factor the equation 8x² - 59x + 51.

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