Factor 25x² + 67x + 42

Here we will show you how to factor the quadratic function 25x² + 67x + 42 using the box method. In other words, we will show you how to factor 25x squared plus 67x plus 42 (25x^2 + 67x + 42) 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 + 67x + 42, like this:

a = 25
b = 67
c = 42

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

1	25x	42
x	25x²	42x
	25x	42

We put 25x² (a) in the bottom left square and 42 (c) in the top right square, like this:

1	25x	42
x	25x²	42x
	25x	42

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 42 (a × c), and add together to equal 67 (b).

More specifically, 25 times 42 is 1050. Therefore, we need to find the two numbers that multiply to equal 1050, and add to equal 67.

? × ? = 1050
? + ? = 67

After looking at this problem, we can see that the two numbers that multiply together to equal 1050, and add together to equal 67, are 25 and 42, as illustrated here:

25 × 42 = 1050
25 + 42 = 67

Now, we can fill in the last two squares in our box with 25x and 42x. Place 25x in the upper left square, and place 42x in the lower right square.

1	25x	42
x	25x²	42x
	25x	42

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 42. The greatest common factor of 25x and 42 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	25x	42
x	25x²	42x
	25x	42

Next, let’s look at the bottom row. We have the terms 25x² and 42x. The greatest common factor of 25x² and 42x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	25x	42
x	25x²	42x
	25x	42

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	42
x	25x²	42x
	25x	42

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

42x ÷ x = 42

You can see this value colored in purple below:

1	25x	42
x	25x²	42x
	25x	42

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 + 67x + 42. 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 + 42)

That’s it! Now you know how to factor the equation 25x² + 67x + 42.

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