Factor 93x² + 64x + 11

Here we will show you how to factor the quadratic function 93x² + 64x + 11 using the box method. In other words, we will show you how to factor 93x squared plus 64x plus 11 (93x^2 + 64x + 11) 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 93x² + 64x + 11, like this:

a = 93
b = 64
c = 11

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

1	31x	11
3x	93x²	33x
	31x	11

We put 93x² (a) in the bottom left square and 11 (c) in the top right square, like this:

1	31x	11
3x	93x²	33x
	31x	11

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 93 times 11 (a × c), and add together to equal 64 (b).

More specifically, 93 times 11 is 1023. Therefore, we need to find the two numbers that multiply to equal 1023, and add to equal 64.

? × ? = 1023
? + ? = 64

After looking at this problem, we can see that the two numbers that multiply together to equal 1023, and add together to equal 64, are 31 and 33, as illustrated here:

31 × 33 = 1023
31 + 33 = 64

Now, we can fill in the last two squares in our box with 31x and 33x. Place 31x in the upper left square, and place 33x in the lower right square.

1	31x	11
3x	93x²	33x
	31x	11

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 31x and 11. The greatest common factor of 31x and 11 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	31x	11
3x	93x²	33x
	31x	11

Next, let’s look at the bottom row. We have the terms 93x² and 33x. The greatest common factor of 93x² and 33x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

1	31x	11
3x	93x²	33x
	31x	11

To find the values below the table, we first divide 93x² by 3x (labeled in blue). This gives us 31x.

93x² ÷ 3x = 31x

You can see this value colored in orange below:

1	31x	11
3x	93x²	33x
	31x	11

Next, we divide 33x by 3x (labeled in blue). This gives us 11.

33x ÷ 3x = 11

You can see this value colored in purple below:

1	31x	11
3x	93x²	33x
	31x	11

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 93x² + 64x + 11. 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 + 1)(31x + 11)

That’s it! Now you know how to factor the equation 93x² + 64x + 11.

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