Factor 17x² + 30x + 13

Here we will show you how to factor the quadratic function 17x² + 30x + 13 using the box method. In other words, we will show you how to factor 17x squared plus 30x plus 13 (17x^2 + 30x + 13) 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 17x² + 30x + 13, like this:

a = 17
b = 30
c = 13

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

13	13x	13
17x	17x²	17x
	x	1

We put 17x² (a) in the bottom left square and 13 (c) in the top right square, like this:

13	13x	13
17x	17x²	17x
	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 17 times 13 (a × c), and add together to equal 30 (b).

More specifically, 17 times 13 is 221. Therefore, we need to find the two numbers that multiply to equal 221, and add to equal 30.

? × ? = 221
? + ? = 30

After looking at this problem, we can see that the two numbers that multiply together to equal 221, and add together to equal 30, are 13 and 17, as illustrated here:

13 × 17 = 221
13 + 17 = 30

Now, we can fill in the last two squares in our box with 13x and 17x. Place 13x in the upper left square, and place 17x in the lower right square.

13	13x	13
17x	17x²	17x
	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 13x and 13. The greatest common factor of 13x and 13 is 13. Therefore, we write 13 to the left of the top row. You can see it here in the color green:

13	13x	13
17x	17x²	17x
	x	1

Next, let’s look at the bottom row. We have the terms 17x² and 17x. The greatest common factor of 17x² and 17x is 17x. Therefore, we write 17x to the left of the bottom row. You can see it here in the color blue:

13	13x	13
17x	17x²	17x
	x	1

To find the values below the table, we first divide 17x² by 17x (labeled in blue). This gives us x.

17x² ÷ 17x = x

You can see this value colored in orange below:

13	13x	13
17x	17x²	17x
	x	1

Next, we divide 17x by 17x (labeled in blue). This gives us 1.

17x ÷ 17x = 1

You can see this value colored in purple below:

13	13x	13
17x	17x²	17x
	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 17x² + 30x + 13. 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:

(17x + 13)(x + 1)

That’s it! Now you know how to factor the equation 17x² + 30x + 13.

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