Factor 32x² - 92x + 51

Here we will show you how to factor the quadratic function 32x² - 92x + 51 using the box method. In other words, we will show you how to factor 32x squared minus 92x plus 51 (32x^2 - 92x + 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 32x² - 92x + 51, like this:

a = 32
b = -92
c = 51

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

-17	-68x	51
8x	32x²	-24x
	4x	-3

We put 32x² (a) in the bottom left square and 51 (c) in the top right square, like this:

-17	-68x	51
8x	32x²	-24x
	4x	-3

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 32 times 51 (a × c), and add together to equal -92 (b).

More specifically, 32 times 51 is 1632. Therefore, we need to find the two numbers that multiply to equal 1632, and add to equal -92.

? × ? = 1632
? + ? = -92

After looking at this problem, we can see that the two numbers that multiply together to equal 1632, and add together to equal -92, are -68 and -24, as illustrated here:

-68 × -24 = 1632
-68 + -24 = -92

Now, we can fill in the last two squares in our box with -68x and -24x. Place -68x in the upper left square, and place -24x in the lower right square.

-17	-68x	51
8x	32x²	-24x
	4x	-3

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 -68x and 51. The greatest common factor of -68x and 51 is -17. Therefore, we write -17 to the left of the top row. You can see it here in the color green:

-17	-68x	51
8x	32x²	-24x
	4x	-3

Next, let’s look at the bottom row. We have the terms 32x² and -24x. The greatest common factor of 32x² and -24x is 8x. Therefore, we write 8x to the left of the bottom row. You can see it here in the color blue:

-17	-68x	51
8x	32x²	-24x
	4x	-3

To find the values below the table, we first divide 32x² by 8x (labeled in blue). This gives us 4x.

32x² ÷ 8x = 4x

You can see this value colored in orange below:

-17	-68x	51
8x	32x²	-24x
	4x	-3

Next, we divide -24x by 8x (labeled in blue). This gives us -3.

-24x ÷ 8x = -3

You can see this value colored in purple below:

-17	-68x	51
8x	32x²	-24x
	4x	-3

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 32x² - 92x + 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 - 17)(4x - 3)

That’s it! Now you know how to factor the equation 32x² - 92x + 51.

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