Factor 14x² + 77x + 98

Here we will show you how to factor the quadratic function 14x² + 77x + 98 using the box method. In other words, we will show you how to factor 14x squared plus 77x plus 98 (14x^2 + 77x + 98) 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 + 77x + 98, like this:

a = 14
b = 77
c = 98

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

14	28x	98
7x	14x²	49x
	2x	7

We put 14x² (a) in the bottom left square and 98 (c) in the top right square, like this:

14	28x	98
7x	14x²	49x
	2x	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 14 times 98 (a × c), and add together to equal 77 (b).

More specifically, 14 times 98 is 1372. Therefore, we need to find the two numbers that multiply to equal 1372, and add to equal 77.

? × ? = 1372
? + ? = 77

After looking at this problem, we can see that the two numbers that multiply together to equal 1372, and add together to equal 77, are 28 and 49, as illustrated here:

28 × 49 = 1372
28 + 49 = 77

Now, we can fill in the last two squares in our box with 28x and 49x. Place 28x in the upper left square, and place 49x in the lower right square.

14	28x	98
7x	14x²	49x
	2x	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 28x and 98. The greatest common factor of 28x and 98 is 14. Therefore, we write 14 to the left of the top row. You can see it here in the color green:

14	28x	98
7x	14x²	49x
	2x	7

Next, let’s look at the bottom row. We have the terms 14x² and 49x. The greatest common factor of 14x² and 49x is 7x. Therefore, we write 7x to the left of the bottom row. You can see it here in the color blue:

14	28x	98
7x	14x²	49x
	2x	7

To find the values below the table, we first divide 14x² by 7x (labeled in blue). This gives us 2x.

14x² ÷ 7x = 2x

You can see this value colored in orange below:

14	28x	98
7x	14x²	49x
	2x	7

Next, we divide 49x by 7x (labeled in blue). This gives us 7.

49x ÷ 7x = 7

You can see this value colored in purple below:

14	28x	98
7x	14x²	49x
	2x	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 14x² + 77x + 98. 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:

(7x + 14)(2x + 7)

That’s it! Now you know how to factor the equation 14x² + 77x + 98.

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