Factor 6x² + 53x + 77

Here we will show you how to factor the quadratic function 6x² + 53x + 77 using the box method. In other words, we will show you how to factor 6x squared plus 53x plus 77 (6x^2 + 53x + 77) 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 6x² + 53x + 77, like this:

a = 6
b = 53
c = 77

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

11	11x	77
6x	6x²	42x
	x	7

We put 6x² (a) in the bottom left square and 77 (c) in the top right square, like this:

11	11x	77
6x	6x²	42x
	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 6 times 77 (a × c), and add together to equal 53 (b).

More specifically, 6 times 77 is 462. Therefore, we need to find the two numbers that multiply to equal 462, and add to equal 53.

? × ? = 462
? + ? = 53

After looking at this problem, we can see that the two numbers that multiply together to equal 462, and add together to equal 53, are 11 and 42, as illustrated here:

11 × 42 = 462
11 + 42 = 53

Now, we can fill in the last two squares in our box with 11x and 42x. Place 11x in the upper left square, and place 42x in the lower right square.

11	11x	77
6x	6x²	42x
	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 11x and 77. The greatest common factor of 11x and 77 is 11. Therefore, we write 11 to the left of the top row. You can see it here in the color green:

11	11x	77
6x	6x²	42x
	x	7

Next, let’s look at the bottom row. We have the terms 6x² and 42x. The greatest common factor of 6x² and 42x is 6x. Therefore, we write 6x to the left of the bottom row. You can see it here in the color blue:

11	11x	77
6x	6x²	42x
	x	7

To find the values below the table, we first divide 6x² by 6x (labeled in blue). This gives us x.

6x² ÷ 6x = x

You can see this value colored in orange below:

11	11x	77
6x	6x²	42x
	x	7

Next, we divide 42x by 6x (labeled in blue). This gives us 7.

42x ÷ 6x = 7

You can see this value colored in purple below:

11	11x	77
6x	6x²	42x
	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 6x² + 53x + 77. 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:

(6x + 11)(x + 7)

That’s it! Now you know how to factor the equation 6x² + 53x + 77.

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