Factor 49x² + 63x + 18

Here we will show you how to factor the quadratic function 49x² + 63x + 18 using the box method. In other words, we will show you how to factor 49x squared plus 63x plus 18 (49x^2 + 63x + 18) 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 49x² + 63x + 18, like this:

a = 49
b = 63
c = 18

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

3	21x	18
7x	49x²	42x
	7x	6

We put 49x² (a) in the bottom left square and 18 (c) in the top right square, like this:

3	21x	18
7x	49x²	42x
	7x	6

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 49 times 18 (a × c), and add together to equal 63 (b).

More specifically, 49 times 18 is 882. Therefore, we need to find the two numbers that multiply to equal 882, and add to equal 63.

? × ? = 882
? + ? = 63

After looking at this problem, we can see that the two numbers that multiply together to equal 882, and add together to equal 63, are 21 and 42, as illustrated here:

21 × 42 = 882
21 + 42 = 63

Now, we can fill in the last two squares in our box with 21x and 42x. Place 21x in the upper left square, and place 42x in the lower right square.

3	21x	18
7x	49x²	42x
	7x	6

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 21x and 18. The greatest common factor of 21x and 18 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	21x	18
7x	49x²	42x
	7x	6

Next, let’s look at the bottom row. We have the terms 49x² and 42x. The greatest common factor of 49x² and 42x is 7x. Therefore, we write 7x to the left of the bottom row. You can see it here in the color blue:

3	21x	18
7x	49x²	42x
	7x	6

To find the values below the table, we first divide 49x² by 7x (labeled in blue). This gives us 7x.

49x² ÷ 7x = 7x

You can see this value colored in orange below:

3	21x	18
7x	49x²	42x
	7x	6

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

42x ÷ 7x = 6

You can see this value colored in purple below:

3	21x	18
7x	49x²	42x
	7x	6

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 49x² + 63x + 18. 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 + 3)(7x + 6)

That’s it! Now you know how to factor the equation 49x² + 63x + 18.

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