Factor 35x² + 48x + 16

Here we will show you how to factor the quadratic function 35x² + 48x + 16 using the box method. In other words, we will show you how to factor 35x squared plus 48x plus 16 (35x^2 + 48x + 16) 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 35x² + 48x + 16, like this:

a = 35
b = 48
c = 16

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

4	20x	16
7x	35x²	28x
	5x	4

We put 35x² (a) in the bottom left square and 16 (c) in the top right square, like this:

4	20x	16
7x	35x²	28x
	5x	4

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 35 times 16 (a × c), and add together to equal 48 (b).

More specifically, 35 times 16 is 560. Therefore, we need to find the two numbers that multiply to equal 560, and add to equal 48.

? × ? = 560
? + ? = 48

After looking at this problem, we can see that the two numbers that multiply together to equal 560, and add together to equal 48, are 20 and 28, as illustrated here:

20 × 28 = 560
20 + 28 = 48

Now, we can fill in the last two squares in our box with 20x and 28x. Place 20x in the upper left square, and place 28x in the lower right square.

4	20x	16
7x	35x²	28x
	5x	4

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 20x and 16. The greatest common factor of 20x and 16 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	20x	16
7x	35x²	28x
	5x	4

Next, let’s look at the bottom row. We have the terms 35x² and 28x. The greatest common factor of 35x² and 28x is 7x. Therefore, we write 7x to the left of the bottom row. You can see it here in the color blue:

4	20x	16
7x	35x²	28x
	5x	4

To find the values below the table, we first divide 35x² by 7x (labeled in blue). This gives us 5x.

35x² ÷ 7x = 5x

You can see this value colored in orange below:

4	20x	16
7x	35x²	28x
	5x	4

Next, we divide 28x by 7x (labeled in blue). This gives us 4.

28x ÷ 7x = 4

You can see this value colored in purple below:

4	20x	16
7x	35x²	28x
	5x	4

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 35x² + 48x + 16. 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 + 4)(5x + 4)

That’s it! Now you know how to factor the equation 35x² + 48x + 16.

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