Factor 31x² - 70x + 39

Here we will show you how to factor the quadratic function 31x² - 70x + 39 using the box method. In other words, we will show you how to factor 31x squared minus 70x plus 39 (31x^2 - 70x + 39) 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 31x² - 70x + 39, like this:

a = 31
b = -70
c = 39

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

-39	-39x	39
31x	31x²	-31x
	x	-1

We put 31x² (a) in the bottom left square and 39 (c) in the top right square, like this:

-39	-39x	39
31x	31x²	-31x
	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 31 times 39 (a × c), and add together to equal -70 (b).

More specifically, 31 times 39 is 1209. Therefore, we need to find the two numbers that multiply to equal 1209, and add to equal -70.

? × ? = 1209
? + ? = -70

After looking at this problem, we can see that the two numbers that multiply together to equal 1209, and add together to equal -70, are -39 and -31, as illustrated here:

-39 × -31 = 1209
-39 + -31 = -70

Now, we can fill in the last two squares in our box with -39x and -31x. Place -39x in the upper left square, and place -31x in the lower right square.

-39	-39x	39
31x	31x²	-31x
	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 -39x and 39. The greatest common factor of -39x and 39 is -39. Therefore, we write -39 to the left of the top row. You can see it here in the color green:

-39	-39x	39
31x	31x²	-31x
	x	-1

Next, let’s look at the bottom row. We have the terms 31x² and -31x. The greatest common factor of 31x² and -31x is 31x. Therefore, we write 31x to the left of the bottom row. You can see it here in the color blue:

-39	-39x	39
31x	31x²	-31x
	x	-1

To find the values below the table, we first divide 31x² by 31x (labeled in blue). This gives us x.

31x² ÷ 31x = x

You can see this value colored in orange below:

-39	-39x	39
31x	31x²	-31x
	x	-1

Next, we divide -31x by 31x (labeled in blue). This gives us -1.

-31x ÷ 31x = -1

You can see this value colored in purple below:

-39	-39x	39
31x	31x²	-31x
	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 31x² - 70x + 39. 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:

(31x - 39)(x - 1)

That’s it! Now you know how to factor the equation 31x² - 70x + 39.

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