Factor 25x² + 88x + 76

Here we will show you how to factor the quadratic function 25x² + 88x + 76 using the box method. In other words, we will show you how to factor 25x squared plus 88x plus 76 (25x^2 + 88x + 76) 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 + 88x + 76, like this:

a = 25
b = 88
c = 76

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

38	38x	76
25x	25x²	50x
	x	2

We put 25x² (a) in the bottom left square and 76 (c) in the top right square, like this:

38	38x	76
25x	25x²	50x
	x	2

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 76 (a × c), and add together to equal 88 (b).

More specifically, 25 times 76 is 1900. Therefore, we need to find the two numbers that multiply to equal 1900, and add to equal 88.

? × ? = 1900
? + ? = 88

After looking at this problem, we can see that the two numbers that multiply together to equal 1900, and add together to equal 88, are 38 and 50, as illustrated here:

38 × 50 = 1900
38 + 50 = 88

Now, we can fill in the last two squares in our box with 38x and 50x. Place 38x in the upper left square, and place 50x in the lower right square.

38	38x	76
25x	25x²	50x
	x	2

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 38x and 76. The greatest common factor of 38x and 76 is 38. Therefore, we write 38 to the left of the top row. You can see it here in the color green:

38	38x	76
25x	25x²	50x
	x	2

Next, let’s look at the bottom row. We have the terms 25x² and 50x. The greatest common factor of 25x² and 50x is 25x. Therefore, we write 25x to the left of the bottom row. You can see it here in the color blue:

38	38x	76
25x	25x²	50x
	x	2

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

25x² ÷ 25x = x

You can see this value colored in orange below:

38	38x	76
25x	25x²	50x
	x	2

Next, we divide 50x by 25x (labeled in blue). This gives us 2.

50x ÷ 25x = 2

You can see this value colored in purple below:

38	38x	76
25x	25x²	50x
	x	2

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 + 88x + 76. 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:

(25x + 38)(x + 2)

That’s it! Now you know how to factor the equation 25x² + 88x + 76.

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