Factor 3x² + 34x + 91

Here we will show you how to factor the quadratic function 3x² + 34x + 91 using the box method. In other words, we will show you how to factor 3x squared plus 34x plus 91 (3x^2 + 34x + 91) 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 3x² + 34x + 91, like this:

a = 3
b = 34
c = 91

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

13	13x	91
3x	3x²	21x
	x	7

We put 3x² (a) in the bottom left square and 91 (c) in the top right square, like this:

13	13x	91
3x	3x²	21x
	x	7

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 3 times 91 (a × c), and add together to equal 34 (b).

More specifically, 3 times 91 is 273. Therefore, we need to find the two numbers that multiply to equal 273, and add to equal 34.

? × ? = 273
? + ? = 34

After looking at this problem, we can see that the two numbers that multiply together to equal 273, and add together to equal 34, are 13 and 21, as illustrated here:

13 × 21 = 273
13 + 21 = 34

Now, we can fill in the last two squares in our box with 13x and 21x. Place 13x in the upper left square, and place 21x in the lower right square.

13	13x	91
3x	3x²	21x
	x	7

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 13x and 91. The greatest common factor of 13x and 91 is 13. Therefore, we write 13 to the left of the top row. You can see it here in the color green:

13	13x	91
3x	3x²	21x
	x	7

Next, let’s look at the bottom row. We have the terms 3x² and 21x. The greatest common factor of 3x² and 21x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

13	13x	91
3x	3x²	21x
	x	7

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

3x² ÷ 3x = x

You can see this value colored in orange below:

13	13x	91
3x	3x²	21x
	x	7

Next, we divide 21x by 3x (labeled in blue). This gives us 7.

21x ÷ 3x = 7

You can see this value colored in purple below:

13	13x	91
3x	3x²	21x
	x	7

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 3x² + 34x + 91. 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:

(3x + 13)(x + 7)

That’s it! Now you know how to factor the equation 3x² + 34x + 91.

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