Factor 20x² + 99x + 63

Here we will show you how to factor the quadratic function 20x² + 99x + 63 using the box method. In other words, we will show you how to factor 20x squared plus 99x plus 63 (20x^2 + 99x + 63) 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 20x² + 99x + 63, like this:

a = 20
b = 99
c = 63

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

3	15x	63
4x	20x²	84x
	5x	21

We put 20x² (a) in the bottom left square and 63 (c) in the top right square, like this:

3	15x	63
4x	20x²	84x
	5x	21

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 20 times 63 (a × c), and add together to equal 99 (b).

More specifically, 20 times 63 is 1260. Therefore, we need to find the two numbers that multiply to equal 1260, and add to equal 99.

? × ? = 1260
? + ? = 99

After looking at this problem, we can see that the two numbers that multiply together to equal 1260, and add together to equal 99, are 15 and 84, as illustrated here:

15 × 84 = 1260
15 + 84 = 99

Now, we can fill in the last two squares in our box with 15x and 84x. Place 15x in the upper left square, and place 84x in the lower right square.

3	15x	63
4x	20x²	84x
	5x	21

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 15x and 63. The greatest common factor of 15x and 63 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	15x	63
4x	20x²	84x
	5x	21

Next, let’s look at the bottom row. We have the terms 20x² and 84x. The greatest common factor of 20x² and 84x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

3	15x	63
4x	20x²	84x
	5x	21

To find the values below the table, we first divide 20x² by 4x (labeled in blue). This gives us 5x.

20x² ÷ 4x = 5x

You can see this value colored in orange below:

3	15x	63
4x	20x²	84x
	5x	21

Next, we divide 84x by 4x (labeled in blue). This gives us 21.

84x ÷ 4x = 21

You can see this value colored in purple below:

3	15x	63
4x	20x²	84x
	5x	21

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 20x² + 99x + 63. 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:

(4x + 3)(5x + 21)

That’s it! Now you know how to factor the equation 20x² + 99x + 63.

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