Factor 15x² + 64x + 68

Here we will show you how to factor the quadratic function 15x² + 64x + 68 using the box method. In other words, we will show you how to factor 15x squared plus 64x plus 68 (15x^2 + 64x + 68) 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 15x² + 64x + 68, like this:

a = 15
b = 64
c = 68

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

2	30x	68
x	15x²	34x
	15x	34

We put 15x² (a) in the bottom left square and 68 (c) in the top right square, like this:

2	30x	68
x	15x²	34x
	15x	34

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 15 times 68 (a × c), and add together to equal 64 (b).

More specifically, 15 times 68 is 1020. Therefore, we need to find the two numbers that multiply to equal 1020, and add to equal 64.

? × ? = 1020
? + ? = 64

After looking at this problem, we can see that the two numbers that multiply together to equal 1020, and add together to equal 64, are 30 and 34, as illustrated here:

30 × 34 = 1020
30 + 34 = 64

Now, we can fill in the last two squares in our box with 30x and 34x. Place 30x in the upper left square, and place 34x in the lower right square.

2	30x	68
x	15x²	34x
	15x	34

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 30x and 68. The greatest common factor of 30x and 68 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	30x	68
x	15x²	34x
	15x	34

Next, let’s look at the bottom row. We have the terms 15x² and 34x. The greatest common factor of 15x² and 34x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

2	30x	68
x	15x²	34x
	15x	34

To find the values below the table, we first divide 15x² by x (labeled in blue). This gives us 15x.

15x² ÷ x = 15x

You can see this value colored in orange below:

2	30x	68
x	15x²	34x
	15x	34

Next, we divide 34x by x (labeled in blue). This gives us 34.

34x ÷ x = 34

You can see this value colored in purple below:

2	30x	68
x	15x²	34x
	15x	34

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 15x² + 64x + 68. 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)(15x + 34)

That’s it! Now you know how to factor the equation 15x² + 64x + 68.

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