Factor 25x² + 81x + 56

Here we will show you how to factor the quadratic function 25x² + 81x + 56 using the box method. In other words, we will show you how to factor 25x squared plus 81x plus 56 (25x^2 + 81x + 56) 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 25x² + 81x + 56, like this:

a = 25
b = 81
c = 56

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

1	25x	56
x	25x²	56x
	25x	56

We put 25x² (a) in the bottom left square and 56 (c) in the top right square, like this:

1	25x	56
x	25x²	56x
	25x	56

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 25 times 56 (a × c), and add together to equal 81 (b).

More specifically, 25 times 56 is 1400. Therefore, we need to find the two numbers that multiply to equal 1400, and add to equal 81.

? × ? = 1400
? + ? = 81

After looking at this problem, we can see that the two numbers that multiply together to equal 1400, and add together to equal 81, are 25 and 56, as illustrated here:

25 × 56 = 1400
25 + 56 = 81

Now, we can fill in the last two squares in our box with 25x and 56x. Place 25x in the upper left square, and place 56x in the lower right square.

1	25x	56
x	25x²	56x
	25x	56

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 25x and 56. The greatest common factor of 25x and 56 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	25x	56
x	25x²	56x
	25x	56

Next, let’s look at the bottom row. We have the terms 25x² and 56x. The greatest common factor of 25x² and 56x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	25x	56
x	25x²	56x
	25x	56

To find the values below the table, we first divide 25x² by x (labeled in blue). This gives us 25x.

25x² ÷ x = 25x

You can see this value colored in orange below:

1	25x	56
x	25x²	56x
	25x	56

Next, we divide 56x by x (labeled in blue). This gives us 56.

56x ÷ x = 56

You can see this value colored in purple below:

1	25x	56
x	25x²	56x
	25x	56

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 25x² + 81x + 56. 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:

(x + 1)(25x + 56)

That’s it! Now you know how to factor the equation 25x² + 81x + 56.

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