Factor 17x² + 72x + 76

Here we will show you how to factor the quadratic function 17x² + 72x + 76 using the box method. In other words, we will show you how to factor 17x squared plus 72x plus 76 (17x^2 + 72x + 76) 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 + 72x + 76, like this:

a = 17
b = 72
c = 76

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

2	34x	76
x	17x²	38x
	17x	38

We put 17x² (a) in the bottom left square and 76 (c) in the top right square, like this:

2	34x	76
x	17x²	38x
	17x	38

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 76 (a × c), and add together to equal 72 (b).

More specifically, 17 times 76 is 1292. Therefore, we need to find the two numbers that multiply to equal 1292, and add to equal 72.

? × ? = 1292
? + ? = 72

After looking at this problem, we can see that the two numbers that multiply together to equal 1292, and add together to equal 72, are 34 and 38, as illustrated here:

34 × 38 = 1292
34 + 38 = 72

Now, we can fill in the last two squares in our box with 34x and 38x. Place 34x in the upper left square, and place 38x in the lower right square.

2	34x	76
x	17x²	38x
	17x	38

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 34x and 76. The greatest common factor of 34x and 76 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	34x	76
x	17x²	38x
	17x	38

Next, let’s look at the bottom row. We have the terms 17x² and 38x. The greatest common factor of 17x² and 38x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

2	34x	76
x	17x²	38x
	17x	38

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

17x² ÷ x = 17x

You can see this value colored in orange below:

2	34x	76
x	17x²	38x
	17x	38

Next, we divide 38x by x (labeled in blue). This gives us 38.

38x ÷ x = 38

You can see this value colored in purple below:

2	34x	76
x	17x²	38x
	17x	38

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 + 72x + 76. 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:

(x + 2)(17x + 38)

That’s it! Now you know how to factor the equation 17x² + 72x + 76.

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