Factor 39x² - 100x + 29

Here we will show you how to factor the quadratic function 39x² - 100x + 29 using the box method. In other words, we will show you how to factor 39x squared minus 100x plus 29 (39x^2 - 100x + 29) 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 39x² - 100x + 29, like this:

a = 39
b = -100
c = 29

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

-29	-87x	29
13x	39x²	-13x
	3x	-1

We put 39x² (a) in the bottom left square and 29 (c) in the top right square, like this:

-29	-87x	29
13x	39x²	-13x
	3x	-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 39 times 29 (a × c), and add together to equal -100 (b).

More specifically, 39 times 29 is 1131. Therefore, we need to find the two numbers that multiply to equal 1131, and add to equal -100.

? × ? = 1131
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 1131, and add together to equal -100, are -87 and -13, as illustrated here:

-87 × -13 = 1131
-87 + -13 = -100

Now, we can fill in the last two squares in our box with -87x and -13x. Place -87x in the upper left square, and place -13x in the lower right square.

-29	-87x	29
13x	39x²	-13x
	3x	-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 -87x and 29. The greatest common factor of -87x and 29 is -29. Therefore, we write -29 to the left of the top row. You can see it here in the color green:

-29	-87x	29
13x	39x²	-13x
	3x	-1

Next, let’s look at the bottom row. We have the terms 39x² and -13x. The greatest common factor of 39x² and -13x is 13x. Therefore, we write 13x to the left of the bottom row. You can see it here in the color blue:

-29	-87x	29
13x	39x²	-13x
	3x	-1

To find the values below the table, we first divide 39x² by 13x (labeled in blue). This gives us 3x.

39x² ÷ 13x = 3x

You can see this value colored in orange below:

-29	-87x	29
13x	39x²	-13x
	3x	-1

Next, we divide -13x by 13x (labeled in blue). This gives us -1.

-13x ÷ 13x = -1

You can see this value colored in purple below:

-29	-87x	29
13x	39x²	-13x
	3x	-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 39x² - 100x + 29. 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:

(13x - 29)(3x - 1)

That’s it! Now you know how to factor the equation 39x² - 100x + 29.

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