Factor 14x² + 83x + 65

Here we will show you how to factor the quadratic function 14x² + 83x + 65 using the box method. In other words, we will show you how to factor 14x squared plus 83x plus 65 (14x^2 + 83x + 65) 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 14x² + 83x + 65, like this:

a = 14
b = 83
c = 65

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

13	13x	65
14x	14x²	70x
	x	5

We put 14x² (a) in the bottom left square and 65 (c) in the top right square, like this:

13	13x	65
14x	14x²	70x
	x	5

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 14 times 65 (a × c), and add together to equal 83 (b).

More specifically, 14 times 65 is 910. Therefore, we need to find the two numbers that multiply to equal 910, and add to equal 83.

? × ? = 910
? + ? = 83

After looking at this problem, we can see that the two numbers that multiply together to equal 910, and add together to equal 83, are 13 and 70, as illustrated here:

13 × 70 = 910
13 + 70 = 83

Now, we can fill in the last two squares in our box with 13x and 70x. Place 13x in the upper left square, and place 70x in the lower right square.

13	13x	65
14x	14x²	70x
	x	5

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 65. The greatest common factor of 13x and 65 is 13. Therefore, we write 13 to the left of the top row. You can see it here in the color green:

13	13x	65
14x	14x²	70x
	x	5

Next, let’s look at the bottom row. We have the terms 14x² and 70x. The greatest common factor of 14x² and 70x is 14x. Therefore, we write 14x to the left of the bottom row. You can see it here in the color blue:

13	13x	65
14x	14x²	70x
	x	5

To find the values below the table, we first divide 14x² by 14x (labeled in blue). This gives us x.

14x² ÷ 14x = x

You can see this value colored in orange below:

13	13x	65
14x	14x²	70x
	x	5

Next, we divide 70x by 14x (labeled in blue). This gives us 5.

70x ÷ 14x = 5

You can see this value colored in purple below:

13	13x	65
14x	14x²	70x
	x	5

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 14x² + 83x + 65. 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:

(14x + 13)(x + 5)

That’s it! Now you know how to factor the equation 14x² + 83x + 65.

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