Factor 45x² + 97x + 52

Here we will show you how to factor the quadratic function 45x² + 97x + 52 using the box method. In other words, we will show you how to factor 45x squared plus 97x plus 52 (45x^2 + 97x + 52) 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 45x² + 97x + 52, like this:

a = 45
b = 97
c = 52

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

1	45x	52
x	45x²	52x
	45x	52

We put 45x² (a) in the bottom left square and 52 (c) in the top right square, like this:

1	45x	52
x	45x²	52x
	45x	52

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 45 times 52 (a × c), and add together to equal 97 (b).

More specifically, 45 times 52 is 2340. Therefore, we need to find the two numbers that multiply to equal 2340, and add to equal 97.

? × ? = 2340
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 2340, and add together to equal 97, are 45 and 52, as illustrated here:

45 × 52 = 2340
45 + 52 = 97

Now, we can fill in the last two squares in our box with 45x and 52x. Place 45x in the upper left square, and place 52x in the lower right square.

1	45x	52
x	45x²	52x
	45x	52

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 45x and 52. The greatest common factor of 45x and 52 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	45x	52
x	45x²	52x
	45x	52

Next, let’s look at the bottom row. We have the terms 45x² and 52x. The greatest common factor of 45x² and 52x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	45x	52
x	45x²	52x
	45x	52

To find the values below the table, we first divide 45x² by x (labeled in blue). This gives us 45x.

45x² ÷ x = 45x

You can see this value colored in orange below:

1	45x	52
x	45x²	52x
	45x	52

Next, we divide 52x by x (labeled in blue). This gives us 52.

52x ÷ x = 52

You can see this value colored in purple below:

1	45x	52
x	45x²	52x
	45x	52

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 45x² + 97x + 52. 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)(45x + 52)

That’s it! Now you know how to factor the equation 45x² + 97x + 52.

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